On the Enumeration of Generalized Menages Numbers for all subsets of , {-2, -1, 0, 1, 2} By Shalosh B. Ekhad -------------------------------------------- If pi[i]-i is never in, {-2}, we have the following Theorem number, 1 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-2} The first, 30, terms (for the sake of Sloane) are: [1, 2, 4, 14, 64, 362, 2428, 18806, 165016, 1616786, 17487988, 206918942, 2657907184, 36828901754, 547499510764, 8691268384262, 146725287298888, 2624698909845026, 49592184973992676, 986871395973226286, 20630087248996393888, 451982388752415571082, 10356214297533070441564, 247684559007061347148694, 6172266130715199881283064, 160003398125042408016496562, 4308222738388267405579704148, 120322098889609280151064426046, 3481032560107154796774876237136, 104198640913126939670550150380186] a(n) satisfies the following homogeneous linear recurrence with polynomial coefficients a(n) = (-1 + n) a(-1 + n) + (-3 + n) a(n - 2) and in Maple input format: a(n) = (-1+n)*a(-1+n)+(-3+n)*a(n-2) Just for fun, the number of such permutations of length, 1000, is: 1483262085505909687439441109273982996255963926238322055383233065506832993498\ 910760925216467061646960546921779103697312808653193293484483031962528900\ 396222235865702548889939198259898064567657239463327196812882978164691788\ 668436886243790340276677188345375972040206823932716971528939985795071158\ 820012614899505363860933549415141331992460352005517658495866602951083809\ 237289928988340345293561417831224870088658136368239334598987590399935741\ 588028625668781193599509950615648114458553306864631246154728768498509816\ 356404715654324031138655735484103994785748442677648542639133778289714271\ 668524801577180231097082101864969104853963451811508571539292757372503938\ 877614880147563173394095994114339016139269606977902147256578875394308450\ 481935385657735877699816453206524535522459341673157674873445018215776695\ 816244172782329638979256218474270538588488198738680174331499277145566751\ 254279877285085588417273155599342381433435975997608570012958968786786397\ 037786701380520323867155497904481150749812718225714085246157453154350105\ 291586289593525808890270334922052316415779657128808242658680970211956201\ 865179056276831082129438798278526584539720847594306163459377986958464117\ 515109058351046668670196963184965241735741653359308703811589638590265395\ 379290077424624882973519203949679774056055603849468194593397314879370497\ 915697401805745557658605977415811511408861986468810472246596221403820535\ 018295592707567091156200118048125498978089737578766604208815031321709458\ 666724196090662922317073186771293428727593470683695627961904829332271787\ 458129287916222355428553574286637393956929342684746988282563141334999026\ 889863798589811767121862718537003430436753922079015523794461624022346879\ 438905883927857780056519175046198413104497252673391414751036278345300642\ 650497559408716102784629297486353156667712943029544483796533813524644585\ 057628952027714554627533121280784968045956694691258451582556090278534884\ 493345505771361691271874180296420772991580391563546140477900280818772755\ 378980382741053301866911786789419179488035588882723148393417547349751246\ 807787073622326114353859165014623666651396288438463420800554891039006968\ 863579232544989393220891395824354826302626115451796498930479137983041424\ 328609851752822688569820774556882888177014845324092581737044531372898812\ 147786069362264578736567759540341465056283919269216203670716289538059597\ 163297242315629008422761892544176079348900863143013459054638934308545678\ 797636963358807222914466717116238560659872635353044636253001538848362120\ 554573877460337099487610472688091148078888282565923759821703392051907443\ 82026547099525624896226379846696807128549686 you will NEVER be able to compute this number via permanents! The asymptotic to order, 10, is a(n) = n! exp(-1) / 1 1 1 1 1 1 1 1 1 \ |1 + 2/n + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---| | 2 3 4 5 6 7 8 9 10| \ n n n n n n n n n / and in Maple input format: a(n) = n!*exp(-1)*(1+2/n+1/n^2+1/n^3+1/n^4+1/n^5+1/n^6+1/n^7+1/n^8+1/n^9+1/n^10 ) -------------------------------------------- If pi[i]-i is never in, {-1}, we have the following Theorem number, 2 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-1} The first, 30, terms (for the sake of Sloane) are: [1, 1, 3, 11, 53, 309, 2119, 16687, 148329, 1468457, 16019531, 190899411, 2467007773, 34361893981, 513137616783, 8178130767479, 138547156531409, 2486151753313617, 47106033220679059, 939765362752547227, 19690321886243846661, 432292066866171724421, 9923922230666898717143, 237760636776394448431551, 5934505493938805432851513, 154068892631103602583645049, 4154153845757163802996059099, 116167945043852116348068366947, 3364864615063302680426807870189, 100833776298063636990123342509997] a(n) satisfies the following homogeneous linear recurrence with polynomial coefficients a(n) = (-1 + n) a(-1 + n) + (n - 2) a(n - 2) and in Maple input format: a(n) = (-1+n)*a(-1+n)+(n-2)*a(n-2) Just for fun, the number of such permutations of length, 1000, is: 1481780303720407494442805307330850202920250340184551450162465379359787059393\ 583072525003063784459224135050316915319804626518492666117737927290134093\ 907508238912792803177016468868300372597367901272716289912064013332394561\ 809400626580949727708356263760690696658273168830231637406199663391357397\ 708542960469381552184936936482046352544935968512621544055696466314632540\ 245775162311262405809812121959969029803014835889049805603854740526559308\ 902693247086316294839771419457659297172563457417364434453463603629715232\ 329491197543262657419862154535987059609204062071196739447882704334306010\ 250013369696052297671477199061005295594559632774914620325016526569703432\ 644464182814051788094616078555867590077765779694380483251663207271368958\ 583859627954024149838250341299113514596841574480909573052290906155509126\ 422252526264418702365953651520183401100250460040900575975924996574711468\ 930067485587106783654813579560206136161029673917389100692796851772896486\ 661215109699229413163081712118576788057392633172456575413234385845437359\ 019820197629804041127375196830084140307296252393010496364570105995096888\ 117747561284052037313080901019609715906945269549178939392761159182453887\ 617261898605281641263905651242410888293095115597173767908744990754862697\ 279312664782429810967479718192002014363931979746097993229965985462578502\ 621081026163206708411345384770448417628140065681208476168431711527782876\ 104532146797323857055910107650464636680546893142779388832766888385181340\ 797808518656329158824485208974341290484970257802680066879464282609115948\ 353936247475706208703703146289919477761251895112187228494309044036864863\ 891808991543461259511843265252297847883288080532641336084189724026420463\ 049440054407620639180198100624499491913662399439099081377325579055545807\ 088573424060520234354406460802713759269046497865234085400734683042471631\ 253421726094394227080125448424687424523889283340506594236462281902823685\ 097049060414501832579436051806485797259345314014454137248648403163731368\ 968625347038978621171060030906756540285111274683725224171322003250956191\ 820911175570857205679271219207470389109009736315595702230605715374456847\ 465991843114485477239735678989648343252890175540316271179072314261955747\ 495437580999552382916598522811819323789660797172871558957064686706347797\ 798972747801141875738126634464751637828628063549810673591594615800585274\ 414559933644675661807607669720233022383460437265322259104334649354905819\ 338439065095813146773616959307173578016873119234294223279309255900178993\ 378068632322769192726976735319034744278206275608308493712724558514559347\ 12504373204147581936247172694794860366236687 you will NEVER be able to compute this number via permanents! The asymptotic to order, 10, is a(n) = n! exp(-1) (1 + 1/n) and in Maple input format: a(n) = n!*exp(-1)*(1+1/n) -------------------------------------------- If pi[i]-i is never in, {0}, we have the following Theorem number, 3 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {0} The first, 30, terms (for the sake of Sloane) are: [0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, 2290792932, 32071101049, 481066515734, 7697064251745, 130850092279664, 2355301661033953, 44750731559645106, 895014631192902121, 18795307255050944540, 413496759611120779881, 9510425471055777937262, 228250211305338670494289, 5706255282633466762357224, 148362637348470135821287825, 4005791208408693667174771274, 112162153835443422680893595673, 3252702461227859257745914274516, 97581073836835777732377428235481] a(n) satisfies the following homogeneous linear recurrence with polynomial coefficients a(n) = (n - 1) a(n - 1) + (n - 1) a(n - 2) and in Maple input format: a(n) = (n-1)*a(n-1)+(n-1)*a(n-2) Just for fun, the number of such permutations of length, 1000, is: 1480300003716690803639166141189660542377872467716834615546918460898888171222\ 360711813189873910548675459590726189130673952565926739378359567722411682\ 225282955956835967209806662206094278319048852420295993918145867464929632\ 177223403177771955752603660100590106551721447382848788617582081310047350\ 358184775693687864320616320161884468076859109403218325729966499814817722\ 523251910400861544265546575384584445357657178710339466137717023503056252\ 650043203882433860978792626830828468703859597819544889563899703925789442\ 886604592950312345074787367168818241367836225845351388059822881452853157\ 092920449246805492179297901159845450144415217557357263061954571997705726\ 917546636177873914180435642912954635442323456238142340910752454816552406\ 177681946008016133704545795503609904692149425055853719332957948207301824\ 597654871393025676689264387133050350749500959081818757218706290284427041\ 888179306280825957697116463097109027133895778139249850841954896876020466\ 195020089609619793369712000118458329727664968204252323090144241603833525\ 494325871758045995132242953876207932374921331061948547816753352642454433\ 684063497786265771541539361657951764142802467082096842550210948234219667\ 949312586019262378885020630611799089203891224372800966941803187567295401\ 877435229552876934033446271920081932431500479266831162067898087375203299\ 321759266896310398013332052717730686941198866814394082086345366161621254\ 849682464332990866189720387263201435245301591551228160672094793591589751\ 046761756899429729095389819155186104380589668134545521358106176432683265\ 088847400075630578125577568721198279481770124987200028465843200836028835\ 056752239304157102409433831420876970912375704827813522561628095930489973\ 076363690716903735444753347277222269644018381058041040336988590465080726\ 362211212847672561792613846955758001267778719146087997403331351690780850\ 403018707387007219860265183241445978545343939401105488747714567335488196\ 900148911502998833745690361445040756502842471542911226022625777386345023\ 944680666372606015155904126779976563721389884798926297873448554696259931\ 889022153417439765913357861346124264844165570744850851379226488885571276\ 189802041073412065174561117871776566686203971568747523655416897364591156\ 339098482517035347569029493318500822966694103069801757199864821884463334\ 464508239561580295442683950514237400428199863686124549042552063736848425\ 988571362282393268539068601119113908474985451813508750353980668686219599\ 738700364731082064708908051255917660356516602631662560718590665234944049\ 328739892430338853873103631687347396881324950657650842869854703810748598\ 52651721482664917019227944750044815550686001 you will NEVER be able to compute this number via permanents! The asymptotic to order, 10, is a(n) = n! exp(-1) and in Maple input format: a(n) = n!*exp(-1) -------------------------------------------- If pi[i]-i is never in, {1}, we have the following Theorem number, 4 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {1} The first, 30, terms (for the sake of Sloane) are: [1, 1, 3, 11, 53, 309, 2119, 16687, 148329, 1468457, 16019531, 190899411, 2467007773, 34361893981, 513137616783, 8178130767479, 138547156531409, 2486151753313617, 47106033220679059, 939765362752547227, 19690321886243846661, 432292066866171724421, 9923922230666898717143, 237760636776394448431551, 5934505493938805432851513, 154068892631103602583645049, 4154153845757163802996059099, 116167945043852116348068366947, 3364864615063302680426807870189, 100833776298063636990123342509997] a(n) satisfies the following homogeneous linear recurrence with polynomial coefficients a(n) = (n - 1) a(n - 1) + (n - 2) a(n - 2) and in Maple input format: a(n) = (n-1)*a(n-1)+(n-2)*a(n-2) Just for fun, the number of such permutations of length, 1000, is: 1481780303720407494442805307330850202920250340184551450162465379359787059393\ 583072525003063784459224135050316915319804626518492666117737927290134093\ 907508238912792803177016468868300372597367901272716289912064013332394561\ 809400626580949727708356263760690696658273168830231637406199663391357397\ 708542960469381552184936936482046352544935968512621544055696466314632540\ 245775162311262405809812121959969029803014835889049805603854740526559308\ 902693247086316294839771419457659297172563457417364434453463603629715232\ 329491197543262657419862154535987059609204062071196739447882704334306010\ 250013369696052297671477199061005295594559632774914620325016526569703432\ 644464182814051788094616078555867590077765779694380483251663207271368958\ 583859627954024149838250341299113514596841574480909573052290906155509126\ 422252526264418702365953651520183401100250460040900575975924996574711468\ 930067485587106783654813579560206136161029673917389100692796851772896486\ 661215109699229413163081712118576788057392633172456575413234385845437359\ 019820197629804041127375196830084140307296252393010496364570105995096888\ 117747561284052037313080901019609715906945269549178939392761159182453887\ 617261898605281641263905651242410888293095115597173767908744990754862697\ 279312664782429810967479718192002014363931979746097993229965985462578502\ 621081026163206708411345384770448417628140065681208476168431711527782876\ 104532146797323857055910107650464636680546893142779388832766888385181340\ 797808518656329158824485208974341290484970257802680066879464282609115948\ 353936247475706208703703146289919477761251895112187228494309044036864863\ 891808991543461259511843265252297847883288080532641336084189724026420463\ 049440054407620639180198100624499491913662399439099081377325579055545807\ 088573424060520234354406460802713759269046497865234085400734683042471631\ 253421726094394227080125448424687424523889283340506594236462281902823685\ 097049060414501832579436051806485797259345314014454137248648403163731368\ 968625347038978621171060030906756540285111274683725224171322003250956191\ 820911175570857205679271219207470389109009736315595702230605715374456847\ 465991843114485477239735678989648343252890175540316271179072314261955747\ 495437580999552382916598522811819323789660797172871558957064686706347797\ 798972747801141875738126634464751637828628063549810673591594615800585274\ 414559933644675661807607669720233022383460437265322259104334649354905819\ 338439065095813146773616959307173578016873119234294223279309255900178993\ 378068632322769192726976735319034744278206275608308493712724558514559347\ 12504373204147581936247172694794860366236687 you will NEVER be able to compute this number via permanents! The asymptotic to order, 10, is a(n) = n! exp(-1) (1 + 1/n) and in Maple input format: a(n) = n!*exp(-1)*(1+1/n) -------------------------------------------- If pi[i]-i is never in, {2}, we have the following Theorem number, 5 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {2} The first, 30, terms (for the sake of Sloane) are: [1, 2, 4, 14, 64, 362, 2428, 18806, 165016, 1616786, 17487988, 206918942, 2657907184, 36828901754, 547499510764, 8691268384262, 146725287298888, 2624698909845026, 49592184973992676, 986871395973226286, 20630087248996393888, 451982388752415571082, 10356214297533070441564, 247684559007061347148694, 6172266130715199881283064, 160003398125042408016496562, 4308222738388267405579704148, 120322098889609280151064426046, 3481032560107154796774876237136, 104198640913126939670550150380186] a(n) satisfies the following homogeneous linear recurrence with polynomial coefficients a(n) = (n - 1) a(n - 1) + (n - 3) a(n - 2) and in Maple input format: a(n) = (n-1)*a(n-1)+(n-3)*a(n-2) Just for fun, the number of such permutations of length, 1000, is: 1483262085505909687439441109273982996255963926238322055383233065506832993498\ 910760925216467061646960546921779103697312808653193293484483031962528900\ 396222235865702548889939198259898064567657239463327196812882978164691788\ 668436886243790340276677188345375972040206823932716971528939985795071158\ 820012614899505363860933549415141331992460352005517658495866602951083809\ 237289928988340345293561417831224870088658136368239334598987590399935741\ 588028625668781193599509950615648114458553306864631246154728768498509816\ 356404715654324031138655735484103994785748442677648542639133778289714271\ 668524801577180231097082101864969104853963451811508571539292757372503938\ 877614880147563173394095994114339016139269606977902147256578875394308450\ 481935385657735877699816453206524535522459341673157674873445018215776695\ 816244172782329638979256218474270538588488198738680174331499277145566751\ 254279877285085588417273155599342381433435975997608570012958968786786397\ 037786701380520323867155497904481150749812718225714085246157453154350105\ 291586289593525808890270334922052316415779657128808242658680970211956201\ 865179056276831082129438798278526584539720847594306163459377986958464117\ 515109058351046668670196963184965241735741653359308703811589638590265395\ 379290077424624882973519203949679774056055603849468194593397314879370497\ 915697401805745557658605977415811511408861986468810472246596221403820535\ 018295592707567091156200118048125498978089737578766604208815031321709458\ 666724196090662922317073186771293428727593470683695627961904829332271787\ 458129287916222355428553574286637393956929342684746988282563141334999026\ 889863798589811767121862718537003430436753922079015523794461624022346879\ 438905883927857780056519175046198413104497252673391414751036278345300642\ 650497559408716102784629297486353156667712943029544483796533813524644585\ 057628952027714554627533121280784968045956694691258451582556090278534884\ 493345505771361691271874180296420772991580391563546140477900280818772755\ 378980382741053301866911786789419179488035588882723148393417547349751246\ 807787073622326114353859165014623666651396288438463420800554891039006968\ 863579232544989393220891395824354826302626115451796498930479137983041424\ 328609851752822688569820774556882888177014845324092581737044531372898812\ 147786069362264578736567759540341465056283919269216203670716289538059597\ 163297242315629008422761892544176079348900863143013459054638934308545678\ 797636963358807222914466717116238560659872635353044636253001538848362120\ 554573877460337099487610472688091148078888282565923759821703392051907443\ 82026547099525624896226379846696807128549686 you will NEVER be able to compute this number via permanents! The asymptotic to order, 10, is a(n) = n! exp(-1) / 1 1 1 1 1 1 1 1 1 \ |1 + 2/n + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---| | 2 3 4 5 6 7 8 9 10| \ n n n n n n n n n / and in Maple input format: a(n) = n!*exp(-1)*(1+2/n+1/n^2+1/n^3+1/n^4+1/n^5+1/n^6+1/n^7+1/n^8+1/n^9+1/n^10 ) -------------------------------------------- If pi[i]-i is never in, {-2, -1}, we have the following Theorem number, 6 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-2, -1} The first, 30, terms (for the sake of Sloane) are: [1, 1, 1, 5, 23, 131, 883, 6859, 60301, 591605, 6405317, 75843233, 974763571, 13512607303, 200949508327, 3190881283415, 53880906258521, 964039575154409, 18217997734199113, 362584510633666621, 7580578211464070863, 166099466140519353035, 3806162403831340850651, 91037404655283210049571, 2268799384077227059257061, 58817651071406021653417309, 1583801150479065295621069069, 44235341139103990750163554649, 1279827199458438844712380743275, 38310926796201013312209078599567] a(n) satisfies the following homogeneous linear recurrence with polynomial coefficients a(n) = (n - 1) a(n - 1) + (n - 3) a(n - 2) + a(n - 3) and in Maple input format: a(n) = (n-1)*a(n-1)+(n-3)*a(n-2)+a(n-3) Just for fun, the number of such permutations of length, 1000, is: 5456613542040578011230311847322588379900166727755766710513090333812162997460\ 787796473657124942988884129842365747721776987844256886704114230091170625\ 462080236901372025922907682964633518444164807834708845419254748227198824\ 936201971957109762461386079456746439587111368260646210505424660972741690\ 126808198768493201429030752816662924327204091320755785530690192804805534\ 999124942497762470835383725364089781202073488218047203833237246571461393\ 182863744070797257011122691150609014147476839610131056079878258467433313\ 818414669653589268479748764427027038766246251957424519885657405422146209\ 876125376783210462964746769185485685307250115213087099430437233650596885\ 767936985956894814153845378480009544630890410855228972714346015000432527\ 588578404799676365520524688548573097766877289142195745147380760708144786\ 583337125115335691135111222595438321706358497426978686813200918304945986\ 530462454255986824668826625958820644248768592559072272055404557909003017\ 396660408149947913984867111838385023162095557742283466972224441358550020\ 801529451083329173417255608010040473162432228917295346973833132612657406\ 245797918051772251377186462710270520003321605153956777691649598632415121\ 994991146390220475413215403699308963374413681833006885760090524185425135\ 420273362883492323573234193274212323419214941576940208525732841156450867\ 658659431612412404847872484889106474801708245411213994203198089465499416\ 185483533267139770311779127276364855391845037107157351068730545382338070\ 516756370037169840165790003662650009648206168367708024252495616176060516\ 886040847110586808023500993883521987549961065449137805267367112322874598\ 833713919774419357374140616281091187121092401634174696636959370438098774\ 731983860045784278314840477044713989981953916161508907911418273287603151\ 325227400991690767354687927383148618561155330647488508696008390758883104\ 437713791513839834030215868238140016268215461923531431483324300698883531\ 116925015205853600628136683667909284183029045264944763900327285202483879\ 412060334382967910925120248075619018185472404380523279731201349611700564\ 064031246384258571786422128411978568144719708805551864832284109550592299\ 967713694535178696237251947395510907367197288425933324968731437647012736\ 088687586159413763608820762283687531779225707048794257364749026804439264\ 263227105881772197020802299538144223046618848498468420298875095833449186\ 378359864247899301586179049605060693420905300126216341214142136100119693\ 949619246655094042829724142939299717676180204888466192102307526145503529\ 292065476663511797610595419899574749571853017126070227712765643207046671\ 1696697993256887063358046935878227791075099 you will NEVER be able to compute this number via permanents! The asymptotic to order, 10, is / 1 1 35 19 5297 49351 a(n) = n! exp(-2) |1 + 2/n + ---- - ---- - ----- - ---- - ------ - ------- | 2 3 4 5 6 7 \ 2 n 6 n 24 n 5 n 720 n 5040 n 127601 16574323 1675555981 \ + -------- + --------- + -----------| 8 9 10| 40320 n 181440 n 3628800 n / and in Maple input format: a(n) = n!*exp(-2)*(1+2/n+1/2/n^2-1/6/n^3-35/24/n^4-19/5/n^5-5297/720/n^6-49351/ 5040/n^7+127601/40320/n^8+16574323/181440/n^9+1675555981/3628800/n^10) -------------------------------------------- If pi[i]-i is never in, {-2, 0}, we have the following Theorem number, 7 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-2, 0} The first, 30, terms (for the sake of Sloane) are: [0, 1, 1, 5, 20, 117, 791, 6205, 55004, 543597, 5922929, 70518905, 910711192, 12678337945, 189252400479, 3015217932073, 51067619064872, 916176426422089, 17355904144773969, 346195850534379613, 7252654441500887308, 159210363453697619309, 3654550890669678498263, 87549316782508730057925, 2185063828548841705316708, 56723791110530358356656789, 1529349972513866287699786913, 42764900021529203662525741233, 1238648373200534037695911445744, 37116572672994522344463392681969] a(n) satisfies the following homogeneous linear recurrence with polynomial coefficients a(n) = (n - 1) a(n - 1) + (n - 1) a(n - 2) + (-n + 2) a(n - 3) + (-n + 2) a(n - 4) - a(n - 5) and in Maple input format: a(n) = (n-1)*a(n-1)+(n-1)*a(n-2)+(-n+2)*a(n-3)+(-n+2)*a(n-4)-a(n-5) Just for fun, the number of such permutations of length, 1000, is: 5451167830856042117052092122553903305542295555332908757571075178216031018599\ 825538846829527474261181730731573910251273870401185402682757927535183037\ 202836429093066122954992588396588867136232949554491015050587974201365810\ 559025947270150875431448219490385053861753989645625182436559578632333233\ 475897979537670149809349626180909509614484224932352196439733856151422295\ 280271997512539341629594007353072704266618402493303815968647815491700078\ 072410939564642839683729698463070782895761260225790690588332971635476644\ 468021193463022434552882542597808153938775955324975929848352578546597815\ 631345207149146691256228384725201320419754980949617318984585830155195114\ 520652563902775842295772767597890905227344386063062120134122986957198054\ 090295352255614667372308255153955082484456825229295836053097915261123394\ 456657507985586922044754241750063506948436811944196244139530811327868950\ 093094971902499143340564898166221089710814615010437008535010634325325198\ 924587615945728730867155585838799798456306971160488427066858595963200084\ 904166501031572809731044680087410790136013730038172163663956067563266584\ 337386859611588420220399959210262757164049546052196437842630713754588388\ 077616610348209398916268817869373706807661378672581205135841285209511955\ 160067025294849290314520493307691035468007709492461323636889029642908589\ 876955840213018512865482916894767720842157342639576907957192774530370805\ 474399229243233138925261065095469754995705017162968274084603131667468234\ 117813379491578463916797515490575299028624646635805777171556243210377232\ 427302564452755989843799968291020274704257946912694979604785618983270379\ 678967118094214835695920106943358444067650767285113273707745106693524084\ 441315912136205316420187358646990336629417625983601596140466329463288573\ 899955319578859072725821498340009936691782323813527901396699352625733212\ 287883654092611873073257176400856312214900822175132065438303257371874823\ 483758093588060985801805041906183993658257008791492162666635642045773454\ 823462641220825890156674972769330175392726777412521633505741432511337858\ 445255525831599677051143198736556676117865452530591884593829935999066255\ 589658856171689588876054942229179353210785437109726180588996160052384360\ 425223842073496678986227531284918561431231884985119545046624237896067461\ 693261894415651093639851824872395248570412765962854492721736669198036152\ 335412921691685302746255181449822401793356188370195448240044964816467347\ 667056759858952721156960581733919017175408060304165113712625096050547784\ 113513114361846080774676300335039301577323057305726967392706775587074375\ 6453848757444538301164528690697506288181101 you will NEVER be able to compute this number via permanents! The asymptotic to order, 10, is / 1 4 89 453 27979 189061 a(n) = n! exp(-2) |1 + 1/n + ---- + ---- + ----- + ----- + ------ + ------- | 2 3 4 5 6 7 \ 2 n 3 n 24 n 40 n 720 n 1260 n 26000057 1103411357 56391087941\ + -------- + ---------- + -----------| 8 9 10| 40320 n 362880 n 3628800 n / and in Maple input format: a(n) = n!*exp(-2)*(1+1/n+1/2/n^2+4/3/n^3+89/24/n^4+453/40/n^5+27979/720/n^6+ 189061/1260/n^7+26000057/40320/n^8+1103411357/362880/n^9+56391087941/3628800/n^ 10) -------------------------------------------- If pi[i]-i is never in, {-2, 1}, we have the following Theorem number, 8 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-2, 1} The first, 30, terms (for the sake of Sloane) are: [1, 1, 2, 6, 26, 142, 933, 7137, 62141, 605736, 6528664, 77047567, 987758755, 13666096695, 202918497170, 3218136429222, 54285755494074, 970462181916142, 18326366326689995, 364522216861069491, 7617176658361956683, 166827540112059575200, 3821378430781738072672, 91370695213290468129805, 2276434523762132173440309, 59000225634292490530090893, 1588350198148158778229871602, 44353250219677362440755557830, 1283001623420211400136666948570, 38399574296414733744600059575534] a(n) satisfies the following homogeneous linear recurrence with polynomial coefficients 2 3 4 a(n) = (-113487060 + 489875035 n - 328747690 n + 89631319 n - 10791351 n 5 + 473343 n ) a(n - 1)/(%1) - 2 2 3 4 (181659907 - 192830050 n + 75932260 n - 11377620 n + 568881 n ) a(n - 2)/ 2 3 (%1) - 1/3 (-1390123998 + 2264825929 n - 1184413563 n + 261828533 n 4 5 - 25942674 n + 946686 n ) a(n - 3)/(%1) - 1/3 (1886859630 - 3581285058 n 5 6 4 2 3 - 13158066 n + 473343 n + 145954789 n + 2459243795 n - 822914094 n ) 2 a(n - 4)/(%1) + 2/3 (-1506214470 + 2130337219 n - 1066138200 n 3 4 5 + 244620650 n - 26055600 n + 1042224 n ) a(n - 5)/(%1) + 1/3 (3423434874 5 6 4 2 - 6135181935 n - 14769171 n + 473343 n + 183961237 n + 3836117198 n 3 - 1159565109 n ) a(n - 6)/(%1) - 1/3 (2156496324 - 4109827325 n 5 6 4 2 3 - 13631409 n + 473343 n + 155517187 n + 2735646128 n - 896573371 n ) 2 a(n - 7)/(%1) + 2/3 (-1470387720 + 2082568219 n - 1061361300 n 3 4 5 + 244620650 n - 26055600 n + 1042224 n ) a(n - 8)/(%1) + 1/3 (6168584550 5 6 4 2 - 8651018642 n - 15242514 n + 473343 n + 198065989 n + 4765484375 n 3 4 - 1324071466 n ) a(n - 9)/(%1) - 1/3 (112828008 - 21391626 n 3 2 5 + 170807573 n - 571698027 n + 688718569 n + 946686 n ) a(n - 10)/(%1) + 2 3 4 2 (157775407 - 188053150 n + 75932260 n - 11377620 n + 568881 n ) 2 3 a(n - 11)/(%1) + (1306062935 n - 962603290 - 618811280 n + 131320279 n 4 5 - 12875799 n + 473343 n ) a(n - 12)/(%1) + 4 3 2 (94655798 - 119152023 n + 473343 n - 8142369 n + 49232170 n ) a(n - 13) ------------------------------------------------------------------------- %1 2 3 4 %1 := 417413568 - 316152677 n + 88966900 n - 10791351 n + 473343 n and in Maple input format: a(n) = (-113487060+489875035*n-328747690*n^2+89631319*n^3-10791351*n^4+473343*n ^5)/(417413568-316152677*n+88966900*n^2-10791351*n^3+473343*n^4)*a(n-1)-2*( 181659907-192830050*n+75932260*n^2-11377620*n^3+568881*n^4)/(417413568-\ 316152677*n+88966900*n^2-10791351*n^3+473343*n^4)*a(n-2)-1/3*(-1390123998+ 2264825929*n-1184413563*n^2+261828533*n^3-25942674*n^4+946686*n^5)/(417413568-\ 316152677*n+88966900*n^2-10791351*n^3+473343*n^4)*a(n-3)-1/3*(1886859630-\ 3581285058*n-13158066*n^5+473343*n^6+145954789*n^4+2459243795*n^2-822914094*n^3 )/(417413568-316152677*n+88966900*n^2-10791351*n^3+473343*n^4)*a(n-4)+2/3*(-\ 1506214470+2130337219*n-1066138200*n^2+244620650*n^3-26055600*n^4+1042224*n^5)/ (417413568-316152677*n+88966900*n^2-10791351*n^3+473343*n^4)*a(n-5)+1/3*( 3423434874-6135181935*n-14769171*n^5+473343*n^6+183961237*n^4+3836117198*n^2-\ 1159565109*n^3)/(417413568-316152677*n+88966900*n^2-10791351*n^3+473343*n^4)*a( n-6)-1/3*(2156496324-4109827325*n-13631409*n^5+473343*n^6+155517187*n^4+ 2735646128*n^2-896573371*n^3)/(417413568-316152677*n+88966900*n^2-10791351*n^3+ 473343*n^4)*a(n-7)+2/3*(-1470387720+2082568219*n-1061361300*n^2+244620650*n^3-\ 26055600*n^4+1042224*n^5)/(417413568-316152677*n+88966900*n^2-10791351*n^3+ 473343*n^4)*a(n-8)+1/3*(6168584550-8651018642*n-15242514*n^5+473343*n^6+ 198065989*n^4+4765484375*n^2-1324071466*n^3)/(417413568-316152677*n+88966900*n^ 2-10791351*n^3+473343*n^4)*a(n-9)-1/3*(112828008-21391626*n^4+170807573*n^3-\ 571698027*n^2+688718569*n+946686*n^5)/(417413568-316152677*n+88966900*n^2-\ 10791351*n^3+473343*n^4)*a(n-10)+2*(157775407-188053150*n+75932260*n^2-11377620 *n^3+568881*n^4)/(417413568-316152677*n+88966900*n^2-10791351*n^3+473343*n^4)*a (n-11)+(1306062935*n-962603290-618811280*n^2+131320279*n^3-12875799*n^4+473343* n^5)/(417413568-316152677*n+88966900*n^2-10791351*n^3+473343*n^4)*a(n-12)+( 94655798-119152023*n+473343*n^4-8142369*n^3+49232170*n^2)/(417413568-316152677* n+88966900*n^2-10791351*n^3+473343*n^4)*a(n-13) Just for fun, the number of such permutations of length, 1000, is: 5456624466252000569460768798583929061984258177552625841074461812877735828328\ 682584994668235323381014952046284789105339540499872546016917414834677601\ 000090541616978923247798423008709305889646703563210989510803629881259407\ 770285424889241929615874476329469152768361963922387377266467329465994590\ 424252589951578871602761366061102062412856635652435237610616340907300824\ 151089177675630291087514928670328405211140313840859123817902004697194730\ 574640149618423715997209501260620509905972439573641791753828091299374030\ 536348599849746495539482519803502290608501320661752909068604429535415977\ 141356589705909335158899363641664078229874024874135466566055261203542080\ 361238999921790007200848214690373682420227814356045809337402078647367071\ 524011755390046240895712832991390349631360487689487273436580391855316597\ 740505535889748650415005336154426095392186617050534720094495154589752783\ 554227862925914962394457356326886000458069457853974456870795666262661473\ 497558758886522692285666016433293819346916255048174363190171116359304103\ 168907599387781845518271148748246676580539055219818024319310037304070072\ 688199322110635685908364813234858722769151018043607480726088469897572347\ 925566747254094091009611050727864833781643439255454188239589259410847463\ 445864044772116473143806513016232123142943096832521097313896464842249837\ 174692757743481661882589161477119520991062212325143019788853484736467830\ 275557583859981478663069685081702299136640310812312436998354378446109250\ 256187206400806462068343211891461680139319255311702223937432811932100389\ 263095474798788946181604111951967060382364103513988446971523336918830983\ 622423273978043432412468465946104365001920397632716058480491925115567261\ 886221879151589574395020409243276732092402561204805567177127406347430945\ 995184354563232891253661200176631861892317876916825671265170747809324228\ 926334085835141445423815805179260283916212409932899507835640588834098551\ 585078718193202683054840252213893954721769214465702505612593635772967504\ 725007880319518554605994624325977130989259025383616492870537664892655294\ 959705656375973697533165949873063829674345433608490103145612712540391125\ 526334584463943044297871358174232796916736623016984978151927968548718405\ 536664709808521978607308579216801631830914254695072992284395764039100892\ 647358416061231736195116779810257742720264872857329826743216213923161612\ 545519769604670055723068306807036847473031873458988621644959174282362081\ 574238213971923606676074178850449663714089519919697889928080808548332045\ 158799327479488575641439482328970674587960390354384322341606563794525838\ 2886466458582304838334436084953483315556966 you will NEVER be able to compute this number via permanents! The asymptotic to order, 10, is / 5 35 397 823 148963 876481 a(n) = n! exp(-2) |1 + 2/n + ---- + ---- + ----- + ----- + ------ + ------- | 2 3 4 5 6 7 \ 2 n 6 n 24 n 15 n 720 n 1008 n 3596597 3651930787 56331867943\ + ------- + ---------- + -----------| 8 9 10 | 896 n 181440 n 518400 n / and in Maple input format: a(n) = n!*exp(-2)*(1+2/n+5/2/n^2+35/6/n^3+397/24/n^4+823/15/n^5+148963/720/n^6+ 876481/1008/n^7+3596597/896/n^8+3651930787/181440/n^9+56331867943/518400/n^10) -------------------------------------------- If pi[i]-i is never in, {-2, 2}, we have the following Theorem number, 9 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-2, 2} The first, 30, terms (for the sake of Sloane) are: [1, 2, 3, 9, 34, 176, 1106, 8241, 70371, 676098, 7204713, 84252233, 1072010712, 14738107136, 217656602456, 3435793029849, 57721548509705, 1028183730411650, 19354550056977555, 383876766917923073, 8001053425278668706, 174828593537337033648, 3996207024319062050994, 95366902237609517126265, 2371801425999741536480435, 61372027060292231912491394, 1649722225208451008166866881, 46002972444885813446946983601, 1329004595865097213556281870176, 39728578892279830958129009438720] a(n) satisfies the following homogeneous linear recurrence with polynomial coefficients 2 3 a(n) = (32286363696 - 290847250340 n + 201294433179 n - 44216918263 n 4 + 3056464578 n ) a(n - 1)/(%1) + 1/2 (966575005790 - 1078246860659 n 2 3 4 + 379322630828 n - 57148614997 n + 3016130134 n ) a(n - 2)/(%1) - 1/2 ( 3 2 -9942457316 + 195141169579 n - 248946053684 n - 100494817408 n 4 5 - 44216918263 n + 3056464578 n ) a(n - 3)/(%1) + 1/2 (866046896404 2 3 4 - 1690223734464 n + 885976469727 n - 172744953686 n + 11020294401 n 5 + 20167222 n ) a(n - 4)/(%1) + 1/2 (166002677020 + 906570187412 n 2 3 4 5 - 988743180861 n + 363520679384 n - 56503278241 n + 3056464578 n ) 2 a(n - 5)/(%1) - 1/2 (2711937783532 - 2531584540894 n + 846966626303 n 3 4 5 - 116166436452 n + 4826696357 n + 20167222 n ) a(n - 6)/(%1) + 1/2 ( 2 3 -564806167324 + 2003640063154 n - 1535368326119 n + 467720837780 n 4 5 - 62575872953 n + 3056464578 n ) a(n - 7)/(%1) - 1/2 (2449489331644 2 3 4 - 2994904962760 n + 1168200853101 n - 190003621288 n + 10899291069 n 5 + 20167222 n ) a(n - 8)/(%1) - 1/2 (-1248137906540 + 3783831537450 n 2 3 4 5 - 2467397921229 n + 653314240030 n - 74842065709 n + 3056464578 n ) 5 4 a(n - 9)/(%1) + 1/2 (3549438160078 + 20167222 n + 1649228447 n 3 2 - 73825026791 n + 763784532251 n - 2878409799659 n) a(n - 10)/(%1) - 1/2 2 3 (1023475870088 - 1417710734518 n + 604958740333 n - 105607394527 n 4 + 6092761934 n ) a(n - 11)/(%1) + (-395486735266 + 261126875921 n 2 3 4 - 51195162186 n + 1830733445 n + 20167222 n ) a(n - 12)/(%1) 2 3 (49813134572 - 31444882181 n + 5068703021 n + 20167222 n ) a(n - 13) + --------------------------------------------------------------------- %1 2 3 %1 := -285170261718 + 200931430357 n - 44237085485 n + 3056464578 n and in Maple input format: a(n) = (32286363696-290847250340*n+201294433179*n^2-44216918263*n^3+3056464578* n^4)/(-285170261718+200931430357*n-44237085485*n^2+3056464578*n^3)*a(n-1)+1/2*( 966575005790-1078246860659*n+379322630828*n^2-57148614997*n^3+3016130134*n^4)/( -285170261718+200931430357*n-44237085485*n^2+3056464578*n^3)*a(n-2)-1/2*(-\ 9942457316+195141169579*n^3-248946053684*n^2-100494817408*n-44216918263*n^4+ 3056464578*n^5)/(-285170261718+200931430357*n-44237085485*n^2+3056464578*n^3)*a (n-3)+1/2*(866046896404-1690223734464*n+885976469727*n^2-172744953686*n^3+ 11020294401*n^4+20167222*n^5)/(-285170261718+200931430357*n-44237085485*n^2+ 3056464578*n^3)*a(n-4)+1/2*(166002677020+906570187412*n-988743180861*n^2+ 363520679384*n^3-56503278241*n^4+3056464578*n^5)/(-285170261718+200931430357*n-\ 44237085485*n^2+3056464578*n^3)*a(n-5)-1/2*(2711937783532-2531584540894*n+ 846966626303*n^2-116166436452*n^3+4826696357*n^4+20167222*n^5)/(-285170261718+ 200931430357*n-44237085485*n^2+3056464578*n^3)*a(n-6)+1/2*(-564806167324+ 2003640063154*n-1535368326119*n^2+467720837780*n^3-62575872953*n^4+3056464578*n ^5)/(-285170261718+200931430357*n-44237085485*n^2+3056464578*n^3)*a(n-7)-1/2*( 2449489331644-2994904962760*n+1168200853101*n^2-190003621288*n^3+10899291069*n^ 4+20167222*n^5)/(-285170261718+200931430357*n-44237085485*n^2+3056464578*n^3)*a (n-8)-1/2*(-1248137906540+3783831537450*n-2467397921229*n^2+653314240030*n^3-\ 74842065709*n^4+3056464578*n^5)/(-285170261718+200931430357*n-44237085485*n^2+ 3056464578*n^3)*a(n-9)+1/2*(3549438160078+20167222*n^5+1649228447*n^4-\ 73825026791*n^3+763784532251*n^2-2878409799659*n)/(-285170261718+200931430357*n -44237085485*n^2+3056464578*n^3)*a(n-10)-1/2*(1023475870088-1417710734518*n+ 604958740333*n^2-105607394527*n^3+6092761934*n^4)/(-285170261718+200931430357*n -44237085485*n^2+3056464578*n^3)*a(n-11)+(-395486735266+261126875921*n-\ 51195162186*n^2+1830733445*n^3+20167222*n^4)/(-285170261718+200931430357*n-\ 44237085485*n^2+3056464578*n^3)*a(n-12)+(49813134572-31444882181*n+5068703021*n ^2+20167222*n^3)/(-285170261718+200931430357*n-44237085485*n^2+3056464578*n^3)* a(n-13) Just for fun, the number of such permutations of length, 1000, is: 5462086569234975707800781110016024610868824620645150127793840636637500144858\ 289603043881201505023664824657877571215010459689379473819554021628037710\ 715799944921756837837939720859770551189027422210719983262186882347866139\ 196265307378698754563163964163196621232193955497857384435511977673134714\ 195643390519125790317107597516944387432106830175695021832070775235714028\ 247967708846251014582749765694443469796412002144575993694023477827209867\ 060515908955964872576899921774154295211631022872624645895794057336343381\ 796380578589100861406506046476407658635791905934755387298427055981246880\ 411429626487342163679459036269355733703798553136444373345376678873822099\ 510537641955966170901707100367292170741714345014252790818304626712318095\ 128717626002542732071763523142391143001343658786976652788310398392621121\ 175651590935647190321189846732393162904378052630860335741671596465758574\ 034589816262968658995293872017566708952544016731657438087760396677527829\ 367053621764244839603738135214622756369150520324504148672768507663196740\ 152845524851633487689640922857033690028126813389491531009403970770858279\ 992858193935529331291023622497159524886654101563388814319675129036661856\ 816083999430770151981968516063223211470982811070997257566601113683328224\ 281971076771480540874643515206585391000221015507574672685981482801091855\ 746232972015712121396041311539626996790520645386904612839101565152120564\ 616848493894906696180460788457834623829306475596524876355650324580135386\ 796697929603249720795834146097868631226801763721372938521688851639104951\ 614390628791195637293676241646732599548744072193068848516880364095755194\ 510355869090583572870067518090414625922733578774772104368916859506139106\ 403397142093165906076992218081002422028760715878988871503817488041317074\ 466093277102486279830139990472097373347733090312321186697764452437136699\ 643285136459974616593809916249884480532822376642442174384537549569075728\ 021736329015981508753279366038924506684312832295292298884248147089001257\ 424650646311139621895593728146445730240597783353966760659633637930496140\ 249688316485338363874118113993045525063908333398332903039006117562015497\ 831573117060973202139637481450502654333702774488220664641059499953399438\ 617423725370991712723771518640796325254192101423004522243434049377615558\ 901532426252338230729540503455979472719283156148979262478601202447196171\ 736130965677044335862600068916518547492013165112715234425606007957350841\ 282220051089525113099646823740229835425773435840491898487598324411250828\ 292331464425932014085462001892416733941536850349335128232992602024665781\ 8487103650504763227154247065069317766393601 you will NEVER be able to compute this number via permanents! The asymptotic to order, 10, is / 11 43 359 19469 474937 2488879 a(n) = n! exp(-2) |1 + 3/n + ---- + ---- + ---- + ------ + ------ + ------- | 2 3 4 5 6 7 \ 2 n 3 n 8 n 120 n 720 n 840 n 195107723 27879707159 527203972397\ + --------- + ----------- + ------------| 8 9 10 | 13440 n 362880 n 1209600 n / and in Maple input format: a(n) = n!*exp(-2)*(1+3/n+11/2/n^2+43/3/n^3+359/8/n^4+19469/120/n^5+474937/720/n ^6+2488879/840/n^7+195107723/13440/n^8+27879707159/362880/n^9+527203972397/ 1209600/n^10) -------------------------------------------- If pi[i]-i is never in, {-1, 0}, we have the following Theorem number, 10 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-1, 0} The first, 30, terms (for the sake of Sloane) are: [0, 0, 1, 3, 16, 96, 675, 5413, 48800, 488592, 5379333, 64595975, 840192288, 11767626752, 176574062535, 2825965531593, 48052401132800, 865108807357216, 16439727718351881, 328839946389605643, 6906458590966507696, 151957709012196732000, 3495340527215980878955, 83894765891839051559661, 2097514511766332975258784, 54538727281981516651340080, 1472626181403335929343130125, 41235550049015337374825954319, 1195883473179004834033385704512, 35877924299793988306767481236224] a(n) satisfies the following homogeneous linear recurrence with polynomial coefficients a(n) = (n - 1) a(n - 1) + (n - 1) a(n - 2) + a(n - 3) and in Maple input format: a(n) = (n-1)*a(n-1)+(n-1)*a(n-2)+a(n-3) Just for fun, the number of such permutations of length, 1000, is: 5445716657568539716376786127516629098007154277792761372775511628662023389080\ 425024144303622656884674772955106122638956063353006039290843186424138279\ 292299907647288824111961857344911212411916103477301586915281935717830926\ 306482349192926697910649723612830138514885829304111381257160654172100763\ 480076782860725482912645993444877447133015643752999237799817344031077949\ 216776091206502340188373063723499623736995384138933438676160348089115726\ 718242738666112217322705070054507219792636552614306337740312463474245454\ 147557313359700444561506777750216903349264090829854983908944697845217997\ 331491081764599066886243114374164891374785780741214768205421601422644978\ 261861024567061653295461414673384981463359272089220686270542400537736929\ 936491298556962363866489362504736854265851891983633133205030611049420959\ 123367753874172248395466133009661534735233423516792203879095205704385958\ 040828540510160963939507751931327936612071781101659669918846072529709133\ 617683159148086984954815854976297875405762187177699861954238442662167963\ 246005657739794465954068429790582169500465754814750413439681336469833673\ 399895376719821525140108789424992557287818955951719420955360489644534186\ 041309419659269173842288238013687494526972842784640090171423810580506280\ 550894081323203264057282032481227032548415076759204823957872119956017816\ 109641860722654760680094796423997569907137699671776942137887975552016139\ 246477007351789915702069884402278021673452111028779244610966065436208483\ 301876065500700701703915179144226971122339534218868720261904572051054923\ 285491017850454218528169824483814229101242930992650582314481132655637858\ 450760367921871712157037083631864170353210646784681961130243064752168413\ 058434087258306604398111745057735234076915797847293664277139617705592752\ 197547836024380292540791244419490162476961381810536310566774444806294614\ 914646719260722951489275296454795325540471786725310000708938610565272450\ 330363730429206745464044978062695038882768707753375026991751487976597696\ 574993834824015172324087418478350884357981902231846035639248903506080154\ 303742722841966359621339695615990826895820782564148772251415637162279500\ 299159452138979073705354684763363314050097619705646469276135782146317415\ 647339780531202694715691352809965277890559990648024668359049752164044538\ 772836114569461167391224699163866935072856756165369194896082977801120553\ 121971858565819290114090120647136655421697388046752523612848505940909683\ 029311176387719820873433927910328293399345574906392889598014966975954467\ 817611389637940928963150701174052356241172754924527504415540850210812132\ 4990218879587910639341682838820267263006101 you will NEVER be able to compute this number via permanents! The asymptotic to order, 10, is / 1 1 3 67 245 1087 12299 a(n) = n! exp(-2) |1 - ---- - ---- + ---- + ----- + ------ + ------- - ------- | 2 3 4 5 6 7 8 \ 2 n 6 n 8 n 60 n 144 n 1680 n 1920 n 2648707 32185541 \ - -------- - ----------| 9 10| 90720 n 403200 n / and in Maple input format: a(n) = n!*exp(-2)*(1-1/2/n^2-1/6/n^3+3/8/n^4+67/60/n^5+245/144/n^6+1087/1680/n^ 7-12299/1920/n^8-2648707/90720/n^9-32185541/403200/n^10) -------------------------------------------- If pi[i]-i is never in, {-1, 1}, we have the following Theorem number, 11 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-1, 1} The first, 30, terms (for the sake of Sloane) are: [1, 1, 2, 5, 21, 117, 792, 6205, 55005, 543597, 5922930, 70518905, 910711193, 12678337945, 189252400480, 3015217932073, 51067619064873, 916176426422089, 17355904144773970, 346195850534379613, 7252654441500887309, 159210363453697619309, 3654550890669678498264, 87549316782508730057925, 2185063828548841705316709, 56723791110530358356656789, 1529349972513866287699786914, 42764900021529203662525741233, 1238648373200534037695911445745, 37116572672994522344463392681969] a(n) satisfies the following homogeneous linear recurrence with polynomial coefficients a(n) = n a(n - 1) + (-n + 2) a(n - 3) - a(n - 4) and in Maple input format: a(n) = n*a(n-1)+(-n+2)*a(n-3)-a(n-4) Just for fun, the number of such permutations of length, 1000, is: 5451167830856042117052092122553903305542295555332908757571075178216031018599\ 825538846829527474261181730731573910251273870401185402682757927535183037\ 202836429093066122954992588396588867136232949554491015050587974201365810\ 559025947270150875431448219490385053861753989645625182436559578632333233\ 475897979537670149809349626180909509614484224932352196439733856151422295\ 280271997512539341629594007353072704266618402493303815968647815491700078\ 072410939564642839683729698463070782895761260225790690588332971635476644\ 468021193463022434552882542597808153938775955324975929848352578546597815\ 631345207149146691256228384725201320419754980949617318984585830155195114\ 520652563902775842295772767597890905227344386063062120134122986957198054\ 090295352255614667372308255153955082484456825229295836053097915261123394\ 456657507985586922044754241750063506948436811944196244139530811327868950\ 093094971902499143340564898166221089710814615010437008535010634325325198\ 924587615945728730867155585838799798456306971160488427066858595963200084\ 904166501031572809731044680087410790136013730038172163663956067563266584\ 337386859611588420220399959210262757164049546052196437842630713754588388\ 077616610348209398916268817869373706807661378672581205135841285209511955\ 160067025294849290314520493307691035468007709492461323636889029642908589\ 876955840213018512865482916894767720842157342639576907957192774530370805\ 474399229243233138925261065095469754995705017162968274084603131667468234\ 117813379491578463916797515490575299028624646635805777171556243210377232\ 427302564452755989843799968291020274704257946912694979604785618983270379\ 678967118094214835695920106943358444067650767285113273707745106693524084\ 441315912136205316420187358646990336629417625983601596140466329463288573\ 899955319578859072725821498340009936691782323813527901396699352625733212\ 287883654092611873073257176400856312214900822175132065438303257371874823\ 483758093588060985801805041906183993658257008791492162666635642045773454\ 823462641220825890156674972769330175392726777412521633505741432511337858\ 445255525831599677051143198736556676117865452530591884593829935999066255\ 589658856171689588876054942229179353210785437109726180588996160052384360\ 425223842073496678986227531284918561431231884985119545046624237896067461\ 693261894415651093639851824872395248570412765962854492721736669198036152\ 335412921691685302746255181449822401793356188370195448240044964816467347\ 667056759858952721156960581733919017175408060304165113712625096050547784\ 113513114361846080774676300335039301577323057305726967392706775587074375\ 6453848757444538301164528690697506288181101 you will NEVER be able to compute this number via permanents! The asymptotic to order, 10, is / 1 4 89 453 27979 189061 a(n) = n! exp(-2) |1 + 1/n + ---- + ---- + ----- + ----- + ------ + ------- | 2 3 4 5 6 7 \ 2 n 3 n 24 n 40 n 720 n 1260 n 26000057 1103411357 56391087941\ + -------- + ---------- + -----------| 8 9 10| 40320 n 362880 n 3628800 n / and in Maple input format: a(n) = n!*exp(-2)*(1+1/n+1/2/n^2+4/3/n^3+89/24/n^4+453/40/n^5+27979/720/n^6+ 189061/1260/n^7+26000057/40320/n^8+1103411357/362880/n^9+56391087941/3628800/n^ 10) -------------------------------------------- If pi[i]-i is never in, {-1, 2}, we have the following Theorem number, 12 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-1, 2} The first, 30, terms (for the sake of Sloane) are: [1, 1, 2, 6, 26, 142, 933, 7137, 62141, 605736, 6528664, 77047567, 987758755, 13666096695, 202918497170, 3218136429222, 54285755494074, 970462181916142, 18326366326689995, 364522216861069491, 7617176658361956683, 166827540112059575200, 3821378430781738072672, 91370695213290468129805, 2276434523762132173440309, 59000225634292490530090893, 1588350198148158778229871602, 44353250219677362440755557830, 1283001623420211400136666948570, 38399574296414733744600059575534] a(n) satisfies the following homogeneous linear recurrence with polynomial coefficients 2 3 4 a(n) = (-113487060 + 489875035 n - 328747690 n + 89631319 n - 10791351 n 5 + 473343 n ) a(n - 1)/(%1) - 2 2 3 4 (181659907 - 192830050 n + 75932260 n - 11377620 n + 568881 n ) a(n - 2)/ 2 3 (%1) - 1/3 (-1390123998 + 2264825929 n - 1184413563 n + 261828533 n 4 5 - 25942674 n + 946686 n ) a(n - 3)/(%1) - 1/3 (1886859630 - 3581285058 n 5 6 4 2 3 - 13158066 n + 473343 n + 145954789 n + 2459243795 n - 822914094 n ) 2 a(n - 4)/(%1) + 2/3 (-1506214470 + 2130337219 n - 1066138200 n 3 4 5 + 244620650 n - 26055600 n + 1042224 n ) a(n - 5)/(%1) + 1/3 (3423434874 5 6 4 2 - 6135181935 n - 14769171 n + 473343 n + 183961237 n + 3836117198 n 3 - 1159565109 n ) a(n - 6)/(%1) - 1/3 (2156496324 - 4109827325 n 5 6 4 2 3 - 13631409 n + 473343 n + 155517187 n + 2735646128 n - 896573371 n ) 2 a(n - 7)/(%1) + 2/3 (-1470387720 + 2082568219 n - 1061361300 n 3 4 5 + 244620650 n - 26055600 n + 1042224 n ) a(n - 8)/(%1) + 1/3 (6168584550 5 6 4 2 - 8651018642 n - 15242514 n + 473343 n + 198065989 n + 4765484375 n 3 4 - 1324071466 n ) a(n - 9)/(%1) - 1/3 (112828008 - 21391626 n 3 2 5 + 170807573 n - 571698027 n + 688718569 n + 946686 n ) a(n - 10)/(%1) + 2 3 4 2 (157775407 - 188053150 n + 75932260 n - 11377620 n + 568881 n ) 2 3 a(n - 11)/(%1) + (1306062935 n - 962603290 - 618811280 n + 131320279 n 4 5 - 12875799 n + 473343 n ) a(n - 12)/(%1) + 4 3 2 (94655798 - 119152023 n + 473343 n - 8142369 n + 49232170 n ) a(n - 13) ------------------------------------------------------------------------- %1 2 3 4 %1 := 417413568 - 316152677 n + 88966900 n - 10791351 n + 473343 n and in Maple input format: a(n) = (-113487060+489875035*n-328747690*n^2+89631319*n^3-10791351*n^4+473343*n ^5)/(417413568-316152677*n+88966900*n^2-10791351*n^3+473343*n^4)*a(n-1)-2*( 181659907-192830050*n+75932260*n^2-11377620*n^3+568881*n^4)/(417413568-\ 316152677*n+88966900*n^2-10791351*n^3+473343*n^4)*a(n-2)-1/3*(-1390123998+ 2264825929*n-1184413563*n^2+261828533*n^3-25942674*n^4+946686*n^5)/(417413568-\ 316152677*n+88966900*n^2-10791351*n^3+473343*n^4)*a(n-3)-1/3*(1886859630-\ 3581285058*n-13158066*n^5+473343*n^6+145954789*n^4+2459243795*n^2-822914094*n^3 )/(417413568-316152677*n+88966900*n^2-10791351*n^3+473343*n^4)*a(n-4)+2/3*(-\ 1506214470+2130337219*n-1066138200*n^2+244620650*n^3-26055600*n^4+1042224*n^5)/ (417413568-316152677*n+88966900*n^2-10791351*n^3+473343*n^4)*a(n-5)+1/3*( 3423434874-6135181935*n-14769171*n^5+473343*n^6+183961237*n^4+3836117198*n^2-\ 1159565109*n^3)/(417413568-316152677*n+88966900*n^2-10791351*n^3+473343*n^4)*a( n-6)-1/3*(2156496324-4109827325*n-13631409*n^5+473343*n^6+155517187*n^4+ 2735646128*n^2-896573371*n^3)/(417413568-316152677*n+88966900*n^2-10791351*n^3+ 473343*n^4)*a(n-7)+2/3*(-1470387720+2082568219*n-1061361300*n^2+244620650*n^3-\ 26055600*n^4+1042224*n^5)/(417413568-316152677*n+88966900*n^2-10791351*n^3+ 473343*n^4)*a(n-8)+1/3*(6168584550-8651018642*n-15242514*n^5+473343*n^6+ 198065989*n^4+4765484375*n^2-1324071466*n^3)/(417413568-316152677*n+88966900*n^ 2-10791351*n^3+473343*n^4)*a(n-9)-1/3*(112828008-21391626*n^4+170807573*n^3-\ 571698027*n^2+688718569*n+946686*n^5)/(417413568-316152677*n+88966900*n^2-\ 10791351*n^3+473343*n^4)*a(n-10)+2*(157775407-188053150*n+75932260*n^2-11377620 *n^3+568881*n^4)/(417413568-316152677*n+88966900*n^2-10791351*n^3+473343*n^4)*a (n-11)+(1306062935*n-962603290-618811280*n^2+131320279*n^3-12875799*n^4+473343* n^5)/(417413568-316152677*n+88966900*n^2-10791351*n^3+473343*n^4)*a(n-12)+( 94655798-119152023*n+473343*n^4-8142369*n^3+49232170*n^2)/(417413568-316152677* n+88966900*n^2-10791351*n^3+473343*n^4)*a(n-13) Just for fun, the number of such permutations of length, 1000, is: 5456624466252000569460768798583929061984258177552625841074461812877735828328\ 682584994668235323381014952046284789105339540499872546016917414834677601\ 000090541616978923247798423008709305889646703563210989510803629881259407\ 770285424889241929615874476329469152768361963922387377266467329465994590\ 424252589951578871602761366061102062412856635652435237610616340907300824\ 151089177675630291087514928670328405211140313840859123817902004697194730\ 574640149618423715997209501260620509905972439573641791753828091299374030\ 536348599849746495539482519803502290608501320661752909068604429535415977\ 141356589705909335158899363641664078229874024874135466566055261203542080\ 361238999921790007200848214690373682420227814356045809337402078647367071\ 524011755390046240895712832991390349631360487689487273436580391855316597\ 740505535889748650415005336154426095392186617050534720094495154589752783\ 554227862925914962394457356326886000458069457853974456870795666262661473\ 497558758886522692285666016433293819346916255048174363190171116359304103\ 168907599387781845518271148748246676580539055219818024319310037304070072\ 688199322110635685908364813234858722769151018043607480726088469897572347\ 925566747254094091009611050727864833781643439255454188239589259410847463\ 445864044772116473143806513016232123142943096832521097313896464842249837\ 174692757743481661882589161477119520991062212325143019788853484736467830\ 275557583859981478663069685081702299136640310812312436998354378446109250\ 256187206400806462068343211891461680139319255311702223937432811932100389\ 263095474798788946181604111951967060382364103513988446971523336918830983\ 622423273978043432412468465946104365001920397632716058480491925115567261\ 886221879151589574395020409243276732092402561204805567177127406347430945\ 995184354563232891253661200176631861892317876916825671265170747809324228\ 926334085835141445423815805179260283916212409932899507835640588834098551\ 585078718193202683054840252213893954721769214465702505612593635772967504\ 725007880319518554605994624325977130989259025383616492870537664892655294\ 959705656375973697533165949873063829674345433608490103145612712540391125\ 526334584463943044297871358174232796916736623016984978151927968548718405\ 536664709808521978607308579216801631830914254695072992284395764039100892\ 647358416061231736195116779810257742720264872857329826743216213923161612\ 545519769604670055723068306807036847473031873458988621644959174282362081\ 574238213971923606676074178850449663714089519919697889928080808548332045\ 158799327479488575641439482328970674587960390354384322341606563794525838\ 2886466458582304838334436084953483315556966 you will NEVER be able to compute this number via permanents! The asymptotic to order, 10, is / 5 35 397 823 148963 876481 a(n) = n! exp(-2) |1 + 2/n + ---- + ---- + ----- + ----- + ------ + ------- | 2 3 4 5 6 7 \ 2 n 6 n 24 n 15 n 720 n 1008 n 3596597 3651930787 56331867943\ + ------- + ---------- + -----------| 8 9 10 | 896 n 181440 n 518400 n / and in Maple input format: a(n) = n!*exp(-2)*(1+2/n+5/2/n^2+35/6/n^3+397/24/n^4+823/15/n^5+148963/720/n^6+ 876481/1008/n^7+3596597/896/n^8+3651930787/181440/n^9+56331867943/518400/n^10) -------------------------------------------- If pi[i]-i is never in, {0, 1}, we have the following Theorem number, 13 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {0, 1} The first, 30, terms (for the sake of Sloane) are: [0, 0, 1, 3, 16, 96, 675, 5413, 48800, 488592, 5379333, 64595975, 840192288, 11767626752, 176574062535, 2825965531593, 48052401132800, 865108807357216, 16439727718351881, 328839946389605643, 6906458590966507696, 151957709012196732000, 3495340527215980878955, 83894765891839051559661, 2097514511766332975258784, 54538727281981516651340080, 1472626181403335929343130125, 41235550049015337374825954319, 1195883473179004834033385704512, 35877924299793988306767481236224] a(n) satisfies the following homogeneous linear recurrence with polynomial coefficients a(n) = (n - 1) a(n - 1) + (n - 1) a(n - 2) + a(n - 3) and in Maple input format: a(n) = (n-1)*a(n-1)+(n-1)*a(n-2)+a(n-3) Just for fun, the number of such permutations of length, 1000, is: 5445716657568539716376786127516629098007154277792761372775511628662023389080\ 425024144303622656884674772955106122638956063353006039290843186424138279\ 292299907647288824111961857344911212411916103477301586915281935717830926\ 306482349192926697910649723612830138514885829304111381257160654172100763\ 480076782860725482912645993444877447133015643752999237799817344031077949\ 216776091206502340188373063723499623736995384138933438676160348089115726\ 718242738666112217322705070054507219792636552614306337740312463474245454\ 147557313359700444561506777750216903349264090829854983908944697845217997\ 331491081764599066886243114374164891374785780741214768205421601422644978\ 261861024567061653295461414673384981463359272089220686270542400537736929\ 936491298556962363866489362504736854265851891983633133205030611049420959\ 123367753874172248395466133009661534735233423516792203879095205704385958\ 040828540510160963939507751931327936612071781101659669918846072529709133\ 617683159148086984954815854976297875405762187177699861954238442662167963\ 246005657739794465954068429790582169500465754814750413439681336469833673\ 399895376719821525140108789424992557287818955951719420955360489644534186\ 041309419659269173842288238013687494526972842784640090171423810580506280\ 550894081323203264057282032481227032548415076759204823957872119956017816\ 109641860722654760680094796423997569907137699671776942137887975552016139\ 246477007351789915702069884402278021673452111028779244610966065436208483\ 301876065500700701703915179144226971122339534218868720261904572051054923\ 285491017850454218528169824483814229101242930992650582314481132655637858\ 450760367921871712157037083631864170353210646784681961130243064752168413\ 058434087258306604398111745057735234076915797847293664277139617705592752\ 197547836024380292540791244419490162476961381810536310566774444806294614\ 914646719260722951489275296454795325540471786725310000708938610565272450\ 330363730429206745464044978062695038882768707753375026991751487976597696\ 574993834824015172324087418478350884357981902231846035639248903506080154\ 303742722841966359621339695615990826895820782564148772251415637162279500\ 299159452138979073705354684763363314050097619705646469276135782146317415\ 647339780531202694715691352809965277890559990648024668359049752164044538\ 772836114569461167391224699163866935072856756165369194896082977801120553\ 121971858565819290114090120647136655421697388046752523612848505940909683\ 029311176387719820873433927910328293399345574906392889598014966975954467\ 817611389637940928963150701174052356241172754924527504415540850210812132\ 4990218879587910639341682838820267263006101 you will NEVER be able to compute this number via permanents! The asymptotic to order, 10, is / 1 1 3 67 245 1087 12299 a(n) = n! exp(-2) |1 - ---- - ---- + ---- + ----- + ------ + ------- - ------- | 2 3 4 5 6 7 8 \ 2 n 6 n 8 n 60 n 144 n 1680 n 1920 n 2648707 32185541 \ - -------- - ----------| 9 10| 90720 n 403200 n / and in Maple input format: a(n) = n!*exp(-2)*(1-1/2/n^2-1/6/n^3+3/8/n^4+67/60/n^5+245/144/n^6+1087/1680/n^ 7-12299/1920/n^8-2648707/90720/n^9-32185541/403200/n^10) -------------------------------------------- If pi[i]-i is never in, {0, 2}, we have the following Theorem number, 14 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {0, 2} The first, 30, terms (for the sake of Sloane) are: [0, 1, 1, 5, 20, 117, 791, 6205, 55004, 543597, 5922929, 70518905, 910711192, 12678337945, 189252400479, 3015217932073, 51067619064872, 916176426422089, 17355904144773969, 346195850534379613, 7252654441500887308, 159210363453697619309, 3654550890669678498263, 87549316782508730057925, 2185063828548841705316708, 56723791110530358356656789, 1529349972513866287699786913, 42764900021529203662525741233, 1238648373200534037695911445744, 37116572672994522344463392681969] a(n) satisfies the following homogeneous linear recurrence with polynomial coefficients a(n) = (n - 1) a(n - 1) + (n - 1) a(n - 2) + (-n + 2) a(n - 3) + (-n + 2) a(n - 4) - a(n - 5) and in Maple input format: a(n) = (n-1)*a(n-1)+(n-1)*a(n-2)+(-n+2)*a(n-3)+(-n+2)*a(n-4)-a(n-5) Just for fun, the number of such permutations of length, 1000, is: 5451167830856042117052092122553903305542295555332908757571075178216031018599\ 825538846829527474261181730731573910251273870401185402682757927535183037\ 202836429093066122954992588396588867136232949554491015050587974201365810\ 559025947270150875431448219490385053861753989645625182436559578632333233\ 475897979537670149809349626180909509614484224932352196439733856151422295\ 280271997512539341629594007353072704266618402493303815968647815491700078\ 072410939564642839683729698463070782895761260225790690588332971635476644\ 468021193463022434552882542597808153938775955324975929848352578546597815\ 631345207149146691256228384725201320419754980949617318984585830155195114\ 520652563902775842295772767597890905227344386063062120134122986957198054\ 090295352255614667372308255153955082484456825229295836053097915261123394\ 456657507985586922044754241750063506948436811944196244139530811327868950\ 093094971902499143340564898166221089710814615010437008535010634325325198\ 924587615945728730867155585838799798456306971160488427066858595963200084\ 904166501031572809731044680087410790136013730038172163663956067563266584\ 337386859611588420220399959210262757164049546052196437842630713754588388\ 077616610348209398916268817869373706807661378672581205135841285209511955\ 160067025294849290314520493307691035468007709492461323636889029642908589\ 876955840213018512865482916894767720842157342639576907957192774530370805\ 474399229243233138925261065095469754995705017162968274084603131667468234\ 117813379491578463916797515490575299028624646635805777171556243210377232\ 427302564452755989843799968291020274704257946912694979604785618983270379\ 678967118094214835695920106943358444067650767285113273707745106693524084\ 441315912136205316420187358646990336629417625983601596140466329463288573\ 899955319578859072725821498340009936691782323813527901396699352625733212\ 287883654092611873073257176400856312214900822175132065438303257371874823\ 483758093588060985801805041906183993658257008791492162666635642045773454\ 823462641220825890156674972769330175392726777412521633505741432511337858\ 445255525831599677051143198736556676117865452530591884593829935999066255\ 589658856171689588876054942229179353210785437109726180588996160052384360\ 425223842073496678986227531284918561431231884985119545046624237896067461\ 693261894415651093639851824872395248570412765962854492721736669198036152\ 335412921691685302746255181449822401793356188370195448240044964816467347\ 667056759858952721156960581733919017175408060304165113712625096050547784\ 113513114361846080774676300335039301577323057305726967392706775587074375\ 6453848757444538301164528690697506288181101 you will NEVER be able to compute this number via permanents! The asymptotic to order, 10, is / 1 4 89 453 27979 189061 a(n) = n! exp(-2) |1 + 1/n + ---- + ---- + ----- + ----- + ------ + ------- | 2 3 4 5 6 7 \ 2 n 3 n 24 n 40 n 720 n 1260 n 26000057 1103411357 56391087941\ + -------- + ---------- + -----------| 8 9 10| 40320 n 362880 n 3628800 n / and in Maple input format: a(n) = n!*exp(-2)*(1+1/n+1/2/n^2+4/3/n^3+89/24/n^4+453/40/n^5+27979/720/n^6+ 189061/1260/n^7+26000057/40320/n^8+1103411357/362880/n^9+56391087941/3628800/n^ 10) -------------------------------------------- If pi[i]-i is never in, {1, 2}, we have the following Theorem number, 15 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {1, 2} The first, 30, terms (for the sake of Sloane) are: [1, 1, 1, 5, 23, 131, 883, 6859, 60301, 591605, 6405317, 75843233, 974763571, 13512607303, 200949508327, 3190881283415, 53880906258521, 964039575154409, 18217997734199113, 362584510633666621, 7580578211464070863, 166099466140519353035, 3806162403831340850651, 91037404655283210049571, 2268799384077227059257061, 58817651071406021653417309, 1583801150479065295621069069, 44235341139103990750163554649, 1279827199458438844712380743275, 38310926796201013312209078599567] a(n) satisfies the following homogeneous linear recurrence with polynomial coefficients a(n) = (n - 1) a(n - 1) + (n - 3) a(n - 2) + a(n - 3) and in Maple input format: a(n) = (n-1)*a(n-1)+(n-3)*a(n-2)+a(n-3) Just for fun, the number of such permutations of length, 1000, is: 5456613542040578011230311847322588379900166727755766710513090333812162997460\ 787796473657124942988884129842365747721776987844256886704114230091170625\ 462080236901372025922907682964633518444164807834708845419254748227198824\ 936201971957109762461386079456746439587111368260646210505424660972741690\ 126808198768493201429030752816662924327204091320755785530690192804805534\ 999124942497762470835383725364089781202073488218047203833237246571461393\ 182863744070797257011122691150609014147476839610131056079878258467433313\ 818414669653589268479748764427027038766246251957424519885657405422146209\ 876125376783210462964746769185485685307250115213087099430437233650596885\ 767936985956894814153845378480009544630890410855228972714346015000432527\ 588578404799676365520524688548573097766877289142195745147380760708144786\ 583337125115335691135111222595438321706358497426978686813200918304945986\ 530462454255986824668826625958820644248768592559072272055404557909003017\ 396660408149947913984867111838385023162095557742283466972224441358550020\ 801529451083329173417255608010040473162432228917295346973833132612657406\ 245797918051772251377186462710270520003321605153956777691649598632415121\ 994991146390220475413215403699308963374413681833006885760090524185425135\ 420273362883492323573234193274212323419214941576940208525732841156450867\ 658659431612412404847872484889106474801708245411213994203198089465499416\ 185483533267139770311779127276364855391845037107157351068730545382338070\ 516756370037169840165790003662650009648206168367708024252495616176060516\ 886040847110586808023500993883521987549961065449137805267367112322874598\ 833713919774419357374140616281091187121092401634174696636959370438098774\ 731983860045784278314840477044713989981953916161508907911418273287603151\ 325227400991690767354687927383148618561155330647488508696008390758883104\ 437713791513839834030215868238140016268215461923531431483324300698883531\ 116925015205853600628136683667909284183029045264944763900327285202483879\ 412060334382967910925120248075619018185472404380523279731201349611700564\ 064031246384258571786422128411978568144719708805551864832284109550592299\ 967713694535178696237251947395510907367197288425933324968731437647012736\ 088687586159413763608820762283687531779225707048794257364749026804439264\ 263227105881772197020802299538144223046618848498468420298875095833449186\ 378359864247899301586179049605060693420905300126216341214142136100119693\ 949619246655094042829724142939299717676180204888466192102307526145503529\ 292065476663511797610595419899574749571853017126070227712765643207046671\ 1696697993256887063358046935878227791075099 you will NEVER be able to compute this number via permanents! The asymptotic to order, 10, is / 1 1 35 19 5297 49351 a(n) = n! exp(-2) |1 + 2/n + ---- - ---- - ----- - ---- - ------ - ------- | 2 3 4 5 6 7 \ 2 n 6 n 24 n 5 n 720 n 5040 n 127601 16574323 1675555981 \ + -------- + --------- + -----------| 8 9 10| 40320 n 181440 n 3628800 n / and in Maple input format: a(n) = n!*exp(-2)*(1+2/n+1/2/n^2-1/6/n^3-35/24/n^4-19/5/n^5-5297/720/n^6-49351/ 5040/n^7+127601/40320/n^8+16574323/181440/n^9+1675555981/3628800/n^10) -------------------------------------------- If pi[i]-i is never in, {-2, -1, 0}, we have the following Theorem number, 16 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-2, -1, 0} The first, 30, terms (for the sake of Sloane) are: [0, 0, 0, 1, 5, 33, 236, 1918, 17440, 175649, 1942171, 23396353, 305055960, 4280721564, 64330087888, 1030831875953, 17545848553729, 316150872317105, 6012076099604308, 120330082937778554, 2528525819886170112, 55657855167451780993, 1280736404605380413303, 30750394025631567131329, 769039694784998460886896, 20001468420258808491667512, 540194118569060356429415712, 15129297070110498161942887009, 438850019810675038935803418941, 13168211505240199345047950457921] a(n) satisfies the following homogeneous linear recurrence with polynomial coefficients 2 (-1 - 8 n + n ) a(n - 1) 4 (n - 6) a(n - 2) a(n) = ------------------------ + ------------------ n - 8 n - 8 2 2 (76 - 33 n + 3 n ) a(n - 3) (48 - 15 n + n ) a(n - 4) - --------------------------- + ------------------------- n - 8 n - 8 2 2 (n - 11 n + 28) a(n - 5) (n - 7) a(n - 7) + --------------------------- + (-n + 6) a(n - 6) - ---------------- n - 8 n - 8 and in Maple input format: a(n) = (-1-8*n+n^2)/(n-8)*a(n-1)+4*(n-6)/(n-8)*a(n-2)-(76-33*n+3*n^2)/(n-8)*a(n -3)+(48-15*n+n^2)/(n-8)*a(n-4)+2*(n^2-11*n+28)/(n-8)*a(n-5)+(-n+6)*a(n-6)-(n-7) /(n-8)*a(n-7) Just for fun, the number of such permutations of length, 1000, is: 2003363189347380095441059218164748868753847578241876252639013607465336441399\ 088480902398950134996547262608677730668664688914491328155389393398582007\ 376912048373427072244520329934246227399443603283223160547133766579846396\ 464305106507481055842147058330406001983660741624052950297320585774152473\ 246607349570586528523876951089149598092316016488912193683196216482402564\ 384296601873886566445074232487376995632939983535610608599169773080117926\ 139410995739718807814999860080616204027043795553108575471753363777205224\ 459062858229278497602695703265525605336617438876071414479525962606160060\ 220004885952100976168380800933445955623698586024993484426216524979834797\ 224397344416191611779413664967143989143238830965180856635881422489265548\ 752467005774122814087582247447658387580454221778924269604684303253449387\ 026754833991959980269693274831177943424983247679655067894366747139271087\ 761207866535001274335356123499573427903095230294173698614457170251782201\ 152199275682052210377676843223638881732381422450866210456292393768765360\ 362883523611032717394642717659666635226757253917090416303156345089816131\ 216745015810273306616765621783521027662225421702414513430052860798607842\ 706303479830402430345613893906045445913409055350803929354755908924444849\ 189354151530163679008845633382131566581256167325554556654010615844173365\ 136896283637212724991208152746445338513142738948361891255691948276331661\ 904947526019699594331421085279765270044834932313468108541849175064302227\ 966180848968391456849733876388074479398981629430935448045069186092666379\ 158687923893030006220625240114553507142912105061818441477113899145264771\ 115193241951031828564798603322508511758538697176229351330700267610253559\ 422417206233819018698859328973529983509439839358314776617782546149004244\ 758093535000562481472769454750266640745952281369215861682328277935239066\ 101075142479804055069733293726668489667930526821808615362384068824787693\ 763342662108940394633966937797455171816759167246752047359146848492569978\ 999097015307908021118381173105919604906459255558897191776139563158733879\ 167742023364444544670829743024973847882451806483006164618586278177396969\ 390862933717267881344191179983475418327523744921888074853023598127609211\ 608867717722625250799207132180871678086233024225949331150616840381356671\ 946443904674392897996965476259610763481505106380097309937020546980232123\ 810033658917468992990688721248354161654072703897303871950694385547310465\ 721398636745342062812319033172194212621712002920342591558670177199956879\ 790509835782313284744186369446103679755750444283454384103521237606095633\ 4171565787362291013216447141953850779768401 you will NEVER be able to compute this number via permanents! The asymptotic to order, 10, is / 5 5 21 53 59 4367 118945 a(n) = n! exp(-3) |1 - ---- - ---- - ---- - ----- + ----- + ------ + ------ | 2 3 4 5 6 7 8 \ 2 n 2 n 8 n 20 n 48 n 112 n 384 n 2311663 1755471197\ + ------- + ----------| 9 10| 1120 n 134400 n / and in Maple input format: a(n) = n!*exp(-3)*(1-5/2/n^2-5/2/n^3-21/8/n^4-53/20/n^5+59/48/n^6+4367/112/n^7+ 118945/384/n^8+2311663/1120/n^9+1755471197/134400/n^10) -------------------------------------------- If pi[i]-i is never in, {-2, -1, 1}, we have the following Theorem number, 17 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-2, -1, 1} The first, 30, terms (for the sake of Sloane) are: [1, 1, 1, 2, 8, 42, 284, 2237, 19922, 197540, 2157535, 25732776, 332761180, 4637190145, 69275824135, 1104430942789, 18715169000033, 335904781086932, 6365646299466792, 127013806136273902, 2661584632871573466, 58440365578574865715, 1341716663194116574864, 32147996753342492876786, 802473236517249699820839, 20834798158276335210635376, 561800403501542590090900488, 15711157815567575417388817289, 455102846850706479139903903773, 13638482079553211038782818730081] -------------------------------------------- If pi[i]-i is never in, {-2, -1, 2}, we have the following Theorem number, 18 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-2, -1, 2} The first, 30, terms (for the sake of Sloane) are: [1, 1, 1, 3, 11, 56, 358, 2682, 23117, 224303, 2411856, 28426208, 364138604, 5035426124, 74741088210, 1185037660531, 19986301060643, 357243080693490, 6745502094474772, 134160547223073340, 2803272437392637695, 61392482953421680283, 1406203125459819395261, 33621641553597506229041, 837633067491420878398087, 21709065853265737042416769, 584418920853887003563292044, 16319070044689039759061114480, 472052459589327361018763299460, 14128092348970047426957709133005] -------------------------------------------- If pi[i]-i is never in, {-2, 0, 1}, we have the following Theorem number, 19 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-2, 0, 1} The first, 30, terms (for the sake of Sloane) are: [0, 0, 1, 2, 6, 36, 250, 1995, 17967, 179853, 1979895, 23772616, 309184500, 4330148153, 64971254146, 1039790713446, 17679994308387, 318293783958812, 6048453005760574, 120984008692021404, 2540935568496074080, 55905779230028563123, 1285937600557898923047, 30864715064784652143355, 771666847913246104346183, 20064470626579763572580048, 541768027895557292996553128, 15170191039012393882583754121, 439953461209529357162106202796, 13199089682038426881647938079524] -------------------------------------------- If pi[i]-i is never in, {-2, 0, 2}, we have the following Theorem number, 20 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-2, 0, 2} The first, 30, terms (for the sake of Sloane) are: [0, 1, 0, 4, 8, 53, 310, 2393, 20780, 203808, 2209580, 26227617, 338000948, 4698272249, 70051698036, 1115088624228, 18872506444260, 338388228380165, 6407372917390218, 127757241262068369, 2675583200619928824, 58718106151286608256, 1347507646886966794872, 32274581268628466397025, 805367783907116968460104, 20903900709693104527466321, 563519641338380985066388936, 15755660453044067069523450180, 456299530844432726590393948928, 13671863378522016280839967735349] -------------------------------------------- If pi[i]-i is never in, {-2, 1, 2}, we have the following Theorem number, 21 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-2, 1, 2} The first, 30, terms (for the sake of Sloane) are: [1, 1, 1, 3, 11, 56, 358, 2682, 23117, 224303, 2411856, 28426208, 364138604, 5035426124, 74741088210, 1185037660531, 19986301060643, 357243080693490, 6745502094474772, 134160547223073340, 2803272437392637695, 61392482953421680283, 1406203125459819395261, 33621641553597506229041, 837633067491420878398087, 21709065853265737042416769, 584418920853887003563292044, 16319070044689039759061114480, 472052459589327361018763299460, 14128092348970047426957709133005] -------------------------------------------- If pi[i]-i is never in, {-1, 0, 1}, we have the following Theorem number, 22 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-1, 0, 1} The first, 30, terms (for the sake of Sloane) are: [0, 0, 0, 1, 4, 29, 206, 1708, 15702, 159737, 1780696, 21599745, 283294740, 3995630216, 60312696452, 970234088153, 16571597074140, 299518677455165, 5711583170669554, 114601867572247060, 2413623459384988298, 53238503492701261201, 1227382998752177970288, 29520591675204638641249, 739465749703940619802632, 19260772868204440943374800, 520903771755774857644956648, 14607652602802668936749637649, 424223076861059084500269322228, 12743466783911832431322487886333] a(n) satisfies the following homogeneous linear recurrence with polynomial coefficients a(n) = n a(n - 1) + 4 a(n - 2) + (9 - 3 n) a(n - 3) + (n - 4) a(n - 4) + (-10 + 2 n) a(n - 5) + (-n + 7) a(n - 6) - a(n - 7) and in Maple input format: a(n) = n*a(n-1)+4*a(n-2)+(9-3*n)*a(n-3)+(n-4)*a(n-4)+(-10+2*n)*a(n-5)+(-n+7)*a( n-6)-a(n-7) Just for fun, the number of such permutations of length, 1000, is: 2001361827523879034028038029797018407548714368895492807906416995048153774646\ 712892490138380800666544234809056922838568436299073183873908666933810707\ 043901975756850294079217224649299260965342087317626213526792547351421074\ 480622493631627874872451031690207085299973724344452602201998083469110280\ 693841325422359588111344607160548381148577248836116027397850664984212607\ 072333425494759332927823282839173304622589434922983620867071952583810107\ 170402743948486875684778083389838515914677253336559310011180226800249668\ 981385963222823923234067941326568978499440251034525495555043748068542194\ 369289172442262386216530176268605153013787689633034327217907674824144952\ 984758160588572636884259285524749374489745509587427763037791706810517045\ 660352807544751916959409730465278630093045882513315284044917617252878118\ 239803346338150511777448625912646415457831935624265434880760742044641972\ 245437549406014700572549690132120789990945007937243685765629258082556963\ 856988350992422646335094488181626301993611412208995736300614100189725337\ 092902331058201986387290931527343646518248449568472969060958741567404027\ 344751442125184626143064301243630392163720962664999253024815475668733241\ 003692026045651805686171107397801077915954904274028302066412942365459869\ 263836564317545968636670089926570636115976182661209005500820496767753023\ 517686658109790242965973636990143995050867476557294634781656466679112247\ 633260696394319657796199544492382727528994246988323240531405373638047039\ 391439897989179189126633793515226294115881215102864197301099466802878445\ 427714294537789475896255981931351654652150619714882353542991044302295543\ 300097107746171845698935196687620687604589216318134938317896915880397578\ 564110619896612499770188292483200677946534874710709126391498364364655586\ 433411414241503967597035903226236290896174082039473747807756769546018060\ 258397907725882928808657597676939594782486904191450040479468190062580688\ 238682011506940558079495155625607611110168001562776375299440449987279727\ 076973841084752286037695058651898754097835274958172272136571093652219097\ 929494274430561902820606682884326356655722542713472750787182129570141767\ 769691482610302212782454446284394693167903442155843433897311551719237568\ 917688417161568460741955618809049888052253013076780022618252305159750678\ 159273164480135141738040533939212837375202750746617383228453780981389424\ 491741334390503717557414660707422399083366828546550249946922060941909738\ 171204986442203774257518953731781710873537858290514558979054822422195252\ 947070933944964510931490100022044402760116389453788954341693753341599537\ 9509602814054345472219084473457612690700201 you will NEVER be able to compute this number via permanents! The asymptotic to order, 10, is / 3 1 3 199 5581 87979 a(n) = n! exp(-3) |1 - 1/n - ---- - ---- + ---- + ----- + ------ + ------ | 2 3 4 5 6 7 \ 2 n n 8 n 40 n 240 n 840 n 6636211 32725313 1673532787\ + -------- + -------- + ----------| 8 9 10| 13440 n 13440 n 134400 n / and in Maple input format: a(n) = n!*exp(-3)*(1-1/n-3/2/n^2-1/n^3+3/8/n^4+199/40/n^5+5581/240/n^6+87979/ 840/n^7+6636211/13440/n^8+32725313/13440/n^9+1673532787/134400/n^10) -------------------------------------------- If pi[i]-i is never in, {-1, 0, 2}, we have the following Theorem number, 23 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-1, 0, 2} The first, 30, terms (for the sake of Sloane) are: [0, 0, 1, 2, 6, 36, 250, 1995, 17967, 179853, 1979895, 23772616, 309184500, 4330148153, 64971254146, 1039790713446, 17679994308387, 318293783958812, 6048453005760574, 120984008692021404, 2540935568496074080, 55905779230028563123, 1285937600557898923047, 30864715064784652143355, 771666847913246104346183, 20064470626579763572580048, 541768027895557292996553128, 15170191039012393882583754121, 439953461209529357162106202796, 13199089682038426881647938079524] -------------------------------------------- If pi[i]-i is never in, {-1, 1, 2}, we have the following Theorem number, 24 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-1, 1, 2} The first, 30, terms (for the sake of Sloane) are: [1, 1, 1, 2, 8, 42, 284, 2237, 19922, 197540, 2157535, 25732776, 332761180, 4637190145, 69275824135, 1104430942789, 18715169000033, 335904781086932, 6365646299466792, 127013806136273902, 2661584632871573466, 58440365578574865715, 1341716663194116574864, 32147996753342492876786, 802473236517249699820839, 20834798158276335210635376, 561800403501542590090900488, 15711157815567575417388817289, 455102846850706479139903903773, 13638482079553211038782818730081] -------------------------------------------- If pi[i]-i is never in, {0, 1, 2}, we have the following Theorem number, 25 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {0, 1, 2} The first, 30, terms (for the sake of Sloane) are: [0, 0, 0, 1, 5, 33, 236, 1918, 17440, 175649, 1942171, 23396353, 305055960, 4280721564, 64330087888, 1030831875953, 17545848553729, 316150872317105, 6012076099604308, 120330082937778554, 2528525819886170112, 55657855167451780993, 1280736404605380413303, 30750394025631567131329, 769039694784998460886896, 20001468420258808491667512, 540194118569060356429415712, 15129297070110498161942887009, 438850019810675038935803418941, 13168211505240199345047950457921] a(n) satisfies the following homogeneous linear recurrence with polynomial coefficients 2 (-1 - 8 n + n ) a(n - 1) 4 (n - 6) a(n - 2) a(n) = ------------------------ + ------------------ n - 8 n - 8 2 2 (76 - 33 n + 3 n ) a(n - 3) (48 - 15 n + n ) a(n - 4) - --------------------------- + ------------------------- n - 8 n - 8 2 2 (n - 11 n + 28) a(n - 5) (n - 7) a(n - 7) + --------------------------- + (-n + 6) a(n - 6) - ---------------- n - 8 n - 8 and in Maple input format: a(n) = (-1-8*n+n^2)/(n-8)*a(n-1)+4*(n-6)/(n-8)*a(n-2)-(76-33*n+3*n^2)/(n-8)*a(n -3)+(48-15*n+n^2)/(n-8)*a(n-4)+2*(n^2-11*n+28)/(n-8)*a(n-5)+(-n+6)*a(n-6)-(n-7) /(n-8)*a(n-7) Just for fun, the number of such permutations of length, 1000, is: 2003363189347380095441059218164748868753847578241876252639013607465336441399\ 088480902398950134996547262608677730668664688914491328155389393398582007\ 376912048373427072244520329934246227399443603283223160547133766579846396\ 464305106507481055842147058330406001983660741624052950297320585774152473\ 246607349570586528523876951089149598092316016488912193683196216482402564\ 384296601873886566445074232487376995632939983535610608599169773080117926\ 139410995739718807814999860080616204027043795553108575471753363777205224\ 459062858229278497602695703265525605336617438876071414479525962606160060\ 220004885952100976168380800933445955623698586024993484426216524979834797\ 224397344416191611779413664967143989143238830965180856635881422489265548\ 752467005774122814087582247447658387580454221778924269604684303253449387\ 026754833991959980269693274831177943424983247679655067894366747139271087\ 761207866535001274335356123499573427903095230294173698614457170251782201\ 152199275682052210377676843223638881732381422450866210456292393768765360\ 362883523611032717394642717659666635226757253917090416303156345089816131\ 216745015810273306616765621783521027662225421702414513430052860798607842\ 706303479830402430345613893906045445913409055350803929354755908924444849\ 189354151530163679008845633382131566581256167325554556654010615844173365\ 136896283637212724991208152746445338513142738948361891255691948276331661\ 904947526019699594331421085279765270044834932313468108541849175064302227\ 966180848968391456849733876388074479398981629430935448045069186092666379\ 158687923893030006220625240114553507142912105061818441477113899145264771\ 115193241951031828564798603322508511758538697176229351330700267610253559\ 422417206233819018698859328973529983509439839358314776617782546149004244\ 758093535000562481472769454750266640745952281369215861682328277935239066\ 101075142479804055069733293726668489667930526821808615362384068824787693\ 763342662108940394633966937797455171816759167246752047359146848492569978\ 999097015307908021118381173105919604906459255558897191776139563158733879\ 167742023364444544670829743024973847882451806483006164618586278177396969\ 390862933717267881344191179983475418327523744921888074853023598127609211\ 608867717722625250799207132180871678086233024225949331150616840381356671\ 946443904674392897996965476259610763481505106380097309937020546980232123\ 810033658917468992990688721248354161654072703897303871950694385547310465\ 721398636745342062812319033172194212621712002920342591558670177199956879\ 790509835782313284744186369446103679755750444283454384103521237606095633\ 4171565787362291013216447141953850779768401 you will NEVER be able to compute this number via permanents! The asymptotic to order, 10, is / 5 5 21 53 59 4367 118945 a(n) = n! exp(-3) |1 - ---- - ---- - ---- - ----- + ----- + ------ + ------ | 2 3 4 5 6 7 8 \ 2 n 2 n 8 n 20 n 48 n 112 n 384 n 2311663 1755471197\ + ------- + ----------| 9 10| 1120 n 134400 n / and in Maple input format: a(n) = n!*exp(-3)*(1-5/2/n^2-5/2/n^3-21/8/n^4-53/20/n^5+59/48/n^6+4367/112/n^7+ 118945/384/n^8+2311663/1120/n^9+1755471197/134400/n^10) -------------------------------------------- If pi[i]-i is never in, {-2, -1, 0, 1}, we have the following Theorem number, 26 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-2, -1, 0, 1} The first, 30, terms (for the sake of Sloane) are: [0, 0, 0, 0, 1, 6, 58, 499, 4814, 50284, 572228, 7050770, 93637691, 1334156612, 20308818956, 329025006637, 5653813150732, 102722614426328, 1967763318700136, 39640921470181124, 837836538203311613, 18539041315706787978, 428620090892592760870, 10335423856779681589087, 259494791758068925623050, 6773325293824353907270116, 183537617074031726596583436, 5156068470899846734809762342, 149983181416590693804807519255, 4512239297919673093774431860264] -------------------------------------------- If pi[i]-i is never in, {-2, -1, 0, 2}, we have the following Theorem number, 27 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-2, -1, 0, 2} The first, 30, terms (for the sake of Sloane) are: [0, 0, 0, 1, 2, 11, 73, 601, 5587, 57302, 642490, 7824968, 102924938, 1454736452, 21993920703, 354251414515, 6056619414217, 109556754939699, 2090544023163796, 41969548156786697, 884329891647103872, 19513841850293770706, 450033555829287484038, 10827241212869657478383, 271282844944138978136980, 7067661931835546722737128, 191181473406519880251780956, 5362226971819686885994823587, 155749678105638327550220464368, 4679305447880401221487462342216] -------------------------------------------- If pi[i]-i is never in, {-2, -1, 1, 2}, we have the following Theorem number, 28 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-2, -1, 1, 2} The first, 30, terms (for the sake of Sloane) are: [1, 1, 1, 2, 5, 17, 95, 713, 6354, 63912, 707962, 8544897, 111579179, 1567451917, 23574470689, 377990865970, 6436860120045, 116026610587249, 2207088285980909, 44185285300762813, 928668475151994016, 20445383067060539744, 470535683784439971120, 11298955195198464216421, 282607458386910882315509, 7350854027100787778208757, 198546238601421062030022893, 5561118246500106772576844226, 161319701881095201851726135105, 4840863973500140028068730052853] -------------------------------------------- If pi[i]-i is never in, {-2, 0, 1, 2}, we have the following Theorem number, 29 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-2, 0, 1, 2} The first, 30, terms (for the sake of Sloane) are: [0, 0, 0, 1, 2, 11, 73, 601, 5587, 57302, 642490, 7824968, 102924938, 1454736452, 21993920703, 354251414515, 6056619414217, 109556754939699, 2090544023163796, 41969548156786697, 884329891647103872, 19513841850293770706, 450033555829287484038, 10827241212869657478383, 271282844944138978136980, 7067661931835546722737128, 191181473406519880251780956, 5362226971819686885994823587, 155749678105638327550220464368, 4679305447880401221487462342216] -------------------------------------------- If pi[i]-i is never in, {-1, 0, 1, 2}, we have the following Theorem number, 30 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-1, 0, 1, 2} The first, 30, terms (for the sake of Sloane) are: [0, 0, 0, 0, 1, 6, 58, 499, 4814, 50284, 572228, 7050770, 93637691, 1334156612, 20308818956, 329025006637, 5653813150732, 102722614426328, 1967763318700136, 39640921470181124, 837836538203311613, 18539041315706787978, 428620090892592760870, 10335423856779681589087, 259494791758068925623050, 6773325293824353907270116, 183537617074031726596583436, 5156068470899846734809762342, 149983181416590693804807519255, 4512239297919673093774431860264] -------------------------------------------- If pi[i]-i is never in, {-2, -1, 0, 1, 2}, we have the following Theorem number, 31 Let a(n) be the number of permutations of length n such that pi[i]-i is NEVER in the set, {-2, -1, 0, 1, 2} The first, 30, terms (for the sake of Sloane) are: [0, 0, 0, 0, 0, 1, 8, 112, 1168, 13365, 159414, 2036488, 27780408, 404351752, 6263006598, 102946702825, 1790795492176, 32880327473840, 635630231970048, 12907624693811937, 274744151265431700, 6117666413618771968, 142238172767973342656, 3447269195991352527456, 86950340973295593197748, 2279086905729732243173353, 61993214084947665345339512, 1747676436765241177149013600, 51002054967878798533404762352, 1538981006780484160540666271701] -------------------------------------------------------- This ends this screen-turning book, that took, 1996.079, secs. to generate.