On the enumerating sequences for permutations that avoid moving in the set S for subsets S of cardinality >=2 that are subsets of, {-2, -1, 0, 1, 2} By Shalosh B. Ekhad Theorem Number, 1, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-2, -1} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 1185207200812005791008969182353359, 37853719929461467346176820313084847, 1246913564693923329821239534567915441, 42322798159913587377623395039629193169, 1478914225227196978860821273273878946705, 53159897135793736566920137840428094424693, 1964081703344459127277770508941762912670087, 74533098337722995686098574476533484952604179, 2903017838051010158390312488568016321149506627, 115977384404188491476735112080293317230989783163] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-2, -1} is the following rational function: 2 2 2 1 - 2 u t + t u - u t ------------------------ 2 2 1 - u - 2 u t + t u and in Maple input form (1-2*u*t+t^2*u^2-u^2*t)/(1-u-2*u*t+t^2*u^2) Just for fun the number of such permutations whose length is, 300, equals: 4169674581987394379335753690149182445147003311719188186235185069466847934433737\ 5831639798679276912293100060380471361312948704098040701396480863465866274579617\ 5207052634345963175661818713861180074322597648922938884581461741411913732584437\ 5224612667508956200879166224825159661364259261475054688364194722940978623029583\ 8091300628802194644347631658006166653016096887082264467713602012799221704274983\ 4091900427881895317774614738877248892615497541586320253830634305137715107823777\ 5496888906028557912383820878067338856088061276797368364927603765765888368293151\ 1393168999695454197313272182145136559985492397970051549040529 A(n), satisfies the following linear recurrence equations with polynomial (of degree <= one!) coefficients A(n) = (-1 + n) A(-1 + n) + (n - 3) A(n - 2) + A(n - 3) In Maple input format: A(n) = (-1+n)*A(-1+n)+(n-3)*A(n-2)+A(n-3) valid for n>=, 4 and with initial values A(1) = 1, A(2) = 1, A(3) = 1 Theorem Number, 2, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-2, 0} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 1149372041412233331905864226858367, 36742703109095429070653115390354193, 1211357350198913128604836746797101584, 41149332884865181911421117272449304145, 1439012992395639805507075645863617098337, 51763244833633160856453088148583723732085, 1913798620836106471408735917556315255340484, 72672501974712918015351798722771617038094117, 2832310897941966980007225878850949603090485079, 113219659806842073521214646736082695560192732685] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-2, 0} is the following rational function: 3 3 2 2 2 1 - u t + u t - u t - t u - ------------------------------------------- 2 2 (u t - 1) (u t + 1) (1 - u - 2 u t + t u ) and in Maple input form -(1-u*t+u^3*t^3-u^2*t-t^2*u^2)/(u*t-1)/(u*t+1)/(1-u-2*u*t+t^2*u^2) Just for fun the number of such permutations whose length is, 300, equals: 4155868021401203062604004621507296703893160248368948489224309166869311081676589\ 9102322209776697817733944719692832631977967330054043882643950284122784627084057\ 4794312077922643757213180997237666859468509227609351574554708157903158665404197\ 5374618782327595878721868188939834444368277911560653505626641678585597433607823\ 0798712219855447659414009863065117822374536884996058699280968965188726782394814\ 9123991529732123448393856410256581268258295344073808699461156706681417360241061\ 6692443664586879553864040175709904941175095904711552205843863070235241197938173\ 8088050474459282105732122533938935032133898387242609910423081 A(n), satisfies the following linear recurrence equations with polynomial (of degree <= one!) coefficients A(n) = (-1 + n) A(-1 + n) + (-1 + n) A(n - 2) + (-n + 2) A(n - 3) + (-n + 2) A(n - 4) - A(n - 5) In Maple input format: A(n) = (-1+n)*A(-1+n)+(-1+n)*A(n-2)+(-n+2)*A(n-3)+(-n+2)*A(n-4)-A(n-5) valid for n>=, 6 and with initial values A(1) = 0, A(2) = 1, A(3) = 1, A(4) = 5, A(5) = 20 Theorem Number, 3, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-2, 1} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 1187771615708648065650464285890305, 37930474724804077136303579675700901, 1249287824923717205741140326472258889, 42398620709788899117162257598915640104, 1481411613105428704624237903462526815512, 53244656446738589561077326052046244624667, 1967043277282845060969813243608361429446247, 74639545251995763076321611966379978397021459, 2906950443193962743083547490817329581416987634, 116126610250036036264298194226900025140699009126] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-2, 1} is the following rational function: 4 4 3 3 2 (u t - 1) (t u - u t - u t - u t + 1) - ----------------------------------------------------- 2 2 6 6 3 3 3 2 (1 - u - 2 u t + t u ) (t u - 2 u t - u t + 1) and in Maple input form -(u*t-1)*(t^4*u^4-u^3*t^3-u^2*t-u*t+1)/(1-u-2*u*t+t^2*u^2)/(t^6*u^6-2*u^3*t^3-u ^3*t^2+1) Just for fun the number of such permutations whose length is, 300, equals: 4169767557035165194488299462816818734523694601263461971732523172988161850011333\ 5300938362529177582827755184578248723264171626555306735744047640325103954201175\ 3110802101607952750348784531420959212542979939514569655807649235968224234194366\ 5141618398625403656857402655120735298065986802957447552137166218064737367557613\ 9163605286516295351356058666628830342996000116733124322613283422663250602861908\ 5315730782843425190219679364779086759799310532729577634263691667041586676618273\ 8866806585778697583137451917628377262281288939260217130719545660433952500941902\ 8253554198781722587294574773081734296082127497898805836940831 Theorem Number, 4, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-2, 2} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 1227500194600927896608187459064672, 39157974919405005032911361299044817, 1288445799843122210774045251500262401, 43687066509632021327936296414150784130, 1525098679615060725952174091305180274451, 54769755126353650287029500034779933496857, 2021813032409198711256842741702217089215570, 76661358284404961787578454706141271283028400, 2983611801478367704871125945486818120277778370, 119110222051514403969169320172350190529465261025] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-2, 2} is the following rational function: 2 2 2 4 4 3 3 1 + 2 t u - 2 u t - u t + t u - 2 u t - ------------------------------------------------------------------------ 2 2 4 4 2 2 2 (u t - 1) (u t + 1) (1 - u - 2 u t + t u ) (t u + 2 t u + u t + 1) and in Maple input form -(1+2*t^2*u^2-2*u*t-u^2*t+t^4*u^4-2*u^3*t^3)/(u*t-1)/(u*t+1)/(1-u-2*u*t+t^2*u^2 )/(t^4*u^4+2*t^2*u^2+u^2*t+1) Just for fun the number of such permutations whose length is, 300, equals: 4183713736985519322842694398948790538057396856465762375905754565334010943614065\ 2299590852488137670099673033318161815491360780906523864879821772784636675255182\ 9588090453724160223057662135871587551162351681363114665304574317496713970023360\ 6625581328894620690760930118955593527932875394004812081825463035864324791019269\ 4545241233033669859367536048183788374608982394086279710683026255756486090365259\ 0453698198133106101616036027604627877158530297511358188312118498555249437397861\ 1069907066822351500542459371795838494025429937777371533585959288333823398352402\ 7037598844846692805168463972292485028589802073131959183409081 Theorem Number, 5, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-1, 0} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 1112255468739238809561400834176399, 35593331067683195738747251163495825, 1174614647089817699534183631406747392, 39937975534666268782816280525652202560, 1397863659510774623595654528591167794193, 50324231841237521050946012502720106633747, 1862035376002473310552282829407731531608400, 70758703353876811543943062805215301782753632, 2759638395967254061991874080128177986052390963, 110387348908900106541207420857231745957102247605] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-1, 0} is the following rational function: u t - 1 - --------------------- 2 2 1 - u - 2 u t + t u and in Maple input form -(u*t-1)/(1-u-2*u*t+t^2*u^2) Just for fun the number of such permutations whose length is, 300, equals: 4142014973553763325305651882582042776765785574918619788057224942572467785537427\ 1913168568151864463934224167921916590119280996068591742052138887307771993664400\ 2531293098141759143203792089947491472261446821659576303712016488788754455757637\ 0272675658049840709735231285255123094183427902413559572505794800591159280112149\ 7430938589536891644455661735351320623082248698637843847173273912375691786992656\ 2522441072867582248518286246461338224297212160291459393963543142299958871844546\ 5438153787585754825206523689216492679936980813214255854573669574794428906725495\ 6229987452106259671602336172298469952525774565460768843719831 A(n), satisfies the following linear recurrence equations with polynomial (of degree <= one!) coefficients A(n) = (-1 + n) A(-1 + n) + (-1 + n) A(n - 2) + A(n - 3) In Maple input format: A(n) = (-1+n)*A(-1+n)+(-1+n)*A(n-2)+A(n-3) valid for n>=, 4 and with initial values A(1) = 0, A(2) = 0, A(3) = 1 Theorem Number, 6, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-1, 1} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 1149372041412233331905864226858368, 36742703109095429070653115390354193, 1211357350198913128604836746797101585, 41149332884865181911421117272449304145, 1439012992395639805507075645863617098338, 51763244833633160856453088148583723732085, 1913798620836106471408735917556315255340485, 72672501974712918015351798722771617038094117, 2832310897941966980007225878850949603090485080, 113219659806842073521214646736082695560192732685] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-1, 1} is the following rational function: u t - 1 - --------------------------------- 2 2 (u t + 1) (1 - u - 2 u t + t u ) and in Maple input form -(u*t-1)/(u*t+1)/(1-u-2*u*t+t^2*u^2) Just for fun the number of such permutations whose length is, 300, equals: 4155868021401203062604004621507296703893160248368948489224309166869311081676589\ 9102322209776697817733944719692832631977967330054043882643950284122784627084057\ 4794312077922643757213180997237666859468509227609351574554708157903158665404197\ 5374618782327595878721868188939834444368277911560653505626641678585597433607823\ 0798712219855447659414009863065117822374536884996058699280968965188726782394814\ 9123991529732123448393856410256581268258295344073808699461156706681417360241061\ 6692443664586879553864040175709904941175095904711552205843863070235241197938173\ 8088050474459282105732122533938935032133898387242609910423081 A(n), satisfies the following linear recurrence equations with polynomial (of degree <= one!) coefficients A(n) = n A(-1 + n) + (-n + 2) A(n - 3) - A(n - 4) In Maple input format: A(n) = n*A(-1+n)+(-n+2)*A(n-3)-A(n-4) valid for n>=, 5 and with initial values A(1) = 1, A(2) = 1, A(3) = 2, A(4) = 5 Theorem Number, 7, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-1, 2} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 1187771615708648065650464285890305, 37930474724804077136303579675700901, 1249287824923717205741140326472258889, 42398620709788899117162257598915640104, 1481411613105428704624237903462526815512, 53244656446738589561077326052046244624667, 1967043277282845060969813243608361429446247, 74639545251995763076321611966379978397021459, 2906950443193962743083547490817329581416987634, 116126610250036036264298194226900025140699009126] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-1, 2} is the following rational function: 4 4 3 3 2 (u t - 1) (u t - u t - u t - u t + 1) - ----------------------------------------------------- 2 2 6 6 3 3 3 2 (1 - u - 2 u t + t u ) (t u - 2 u t - u t + 1) and in Maple input form -(u*t-1)*(u^4*t^4-u^3*t^3-u^2*t-u*t+1)/(1-u-2*u*t+t^2*u^2)/(t^6*u^6-2*u^3*t^3-u ^3*t^2+1) Just for fun the number of such permutations whose length is, 300, equals: 4169767557035165194488299462816818734523694601263461971732523172988161850011333\ 5300938362529177582827755184578248723264171626555306735744047640325103954201175\ 3110802101607952750348784531420959212542979939514569655807649235968224234194366\ 5141618398625403656857402655120735298065986802957447552137166218064737367557613\ 9163605286516295351356058666628830342996000116733124322613283422663250602861908\ 5315730782843425190219679364779086759799310532729577634263691667041586676618273\ 8866806585778697583137451917628377262281288939260217130719545660433952500941902\ 8253554198781722587294574773081734296082127497898805836940831 Theorem Number, 8, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {0, 1} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 1112255468739238809561400834176399, 35593331067683195738747251163495825, 1174614647089817699534183631406747392, 39937975534666268782816280525652202560, 1397863659510774623595654528591167794193, 50324231841237521050946012502720106633747, 1862035376002473310552282829407731531608400, 70758703353876811543943062805215301782753632, 2759638395967254061991874080128177986052390963, 110387348908900106541207420857231745957102247605] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {0, 1} is the following rational function: u t - 1 - --------------------- 2 2 1 - u - 2 u t + t u and in Maple input form -(u*t-1)/(1-u-2*u*t+t^2*u^2) Just for fun the number of such permutations whose length is, 300, equals: 4142014973553763325305651882582042776765785574918619788057224942572467785537427\ 1913168568151864463934224167921916590119280996068591742052138887307771993664400\ 2531293098141759143203792089947491472261446821659576303712016488788754455757637\ 0272675658049840709735231285255123094183427902413559572505794800591159280112149\ 7430938589536891644455661735351320623082248698637843847173273912375691786992656\ 2522441072867582248518286246461338224297212160291459393963543142299958871844546\ 5438153787585754825206523689216492679936980813214255854573669574794428906725495\ 6229987452106259671602336172298469952525774565460768843719831 A(n), satisfies the following linear recurrence equations with polynomial (of degree <= one!) coefficients A(n) = (-1 + n) A(-1 + n) + (-1 + n) A(n - 2) + A(n - 3) In Maple input format: A(n) = (-1+n)*A(-1+n)+(-1+n)*A(n-2)+A(n-3) valid for n>=, 4 and with initial values A(1) = 0, A(2) = 0, A(3) = 1 Theorem Number, 9, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {0, 2} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 1149372041412233331905864226858367, 36742703109095429070653115390354193, 1211357350198913128604836746797101584, 41149332884865181911421117272449304145, 1439012992395639805507075645863617098337, 51763244833633160856453088148583723732085, 1913798620836106471408735917556315255340484, 72672501974712918015351798722771617038094117, 2832310897941966980007225878850949603090485079, 113219659806842073521214646736082695560192732685] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {0, 2} is the following rational function: 3 3 2 2 2 1 - u t + u t - u t - t u - ------------------------------------------- 2 2 (u t + 1) (u t - 1) (1 - u - 2 u t + t u ) and in Maple input form -(1-u*t+u^3*t^3-u^2*t-t^2*u^2)/(u*t+1)/(u*t-1)/(1-u-2*u*t+t^2*u^2) Just for fun the number of such permutations whose length is, 300, equals: 4155868021401203062604004621507296703893160248368948489224309166869311081676589\ 9102322209776697817733944719692832631977967330054043882643950284122784627084057\ 4794312077922643757213180997237666859468509227609351574554708157903158665404197\ 5374618782327595878721868188939834444368277911560653505626641678585597433607823\ 0798712219855447659414009863065117822374536884996058699280968965188726782394814\ 9123991529732123448393856410256581268258295344073808699461156706681417360241061\ 6692443664586879553864040175709904941175095904711552205843863070235241197938173\ 8088050474459282105732122533938935032133898387242609910423081 A(n), satisfies the following linear recurrence equations with polynomial (of degree <= one!) coefficients A(n) = (-1 + n) A(-1 + n) + (-1 + n) A(n - 2) + (-n + 2) A(n - 3) + (-n + 2) A(n - 4) - A(n - 5) In Maple input format: A(n) = (-1+n)*A(-1+n)+(-1+n)*A(n-2)+(-n+2)*A(n-3)+(-n+2)*A(n-4)-A(n-5) valid for n>=, 6 and with initial values A(1) = 0, A(2) = 1, A(3) = 1, A(4) = 5, A(5) = 20 Theorem Number, 10, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {1, 2} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 1185207200812005791008969182353359, 37853719929461467346176820313084847, 1246913564693923329821239534567915441, 42322798159913587377623395039629193169, 1478914225227196978860821273273878946705, 53159897135793736566920137840428094424693, 1964081703344459127277770508941762912670087, 74533098337722995686098574476533484952604179, 2903017838051010158390312488568016321149506627, 115977384404188491476735112080293317230989783163] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {1, 2} is the following rational function: 2 2 2 1 - 2 u t + t u - u t ------------------------ 2 2 1 - u - 2 u t + t u and in Maple input form (1-2*u*t+t^2*u^2-u^2*t)/(1-u-2*u*t+t^2*u^2) Just for fun the number of such permutations whose length is, 300, equals: 4169674581987394379335753690149182445147003311719188186235185069466847934433737\ 5831639798679276912293100060380471361312948704098040701396480863465866274579617\ 5207052634345963175661818713861180074322597648922938884581461741411913732584437\ 5224612667508956200879166224825159661364259261475054688364194722940978623029583\ 8091300628802194644347631658006166653016096887082264467713602012799221704274983\ 4091900427881895317774614738877248892615497541586320253830634305137715107823777\ 5496888906028557912383820878067338856088061276797368364927603765765888368293151\ 1393168999695454197313272182145136559985492397970051549040529 A(n), satisfies the following linear recurrence equations with polynomial (of degree <= one!) coefficients A(n) = (-1 + n) A(-1 + n) + (n - 3) A(n - 2) + A(n - 3) In Maple input format: A(n) = (-1+n)*A(-1+n)+(n-3)*A(n-2)+A(n-3) valid for n>=, 4 and with initial values A(1) = 1, A(2) = 1, A(3) = 1 Theorem Number, 11, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-2, -1, 0} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 408290461702265033886375209600444, 13067496008176147709565436787797750, 431293404936392435759266629619185952, 14666022894638868659882008072210107041, 513376309078215882204904416738197847187 , 18483708874174278938151472221941332456129, 683970727473640876057925304995985455940168, 25993460104389728781062433893888150534647764, 1013837552339715632692424896851506601268719472, 40556928588998185722745225030068193448329700177] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-2, -1, 0} is the following rational function: 3 3 2 1 - 2 u t + u t - u t - ---------------------------------------- 3 3 2 (u t - 1) (u t - u t - 2 u t + 1 - u) and in Maple input form -(1-2*u*t+u^3*t^3-u^2*t)/(u*t-1)/(u^3*t^3-u^2*t-2*u*t+1-u) Just for fun the number of such permutations whose length is, 300, equals: 1523728159974578560804352007539109349566119243749248274203537797374262660830053\ 2064436831429958063604578650914884795003622733498737254873057751131683182325768\ 2079350719657933094703245088497668917797021685461913513431577963660173225053177\ 6824958605876926669900237381412052109620646439827131097160454997768956315317714\ 1242101419322760708876080910952444032415129558785956299837128659915971162226465\ 9577709685200164591844737707980003108499553386583327238127950873326416070806663\ 7467259832027592930245878809132793879178387027080651139641037245406609602674517\ 4651814087517012079766463334498321856063729632553498507083521 A(n), satisfies the following linear recurrence equations with polynomial (of degree <= one!) coefficients A(n) = (-1 + n) A(-1 + n) + (2 + n) A(n - 2) + (13 - 3 n) A(n - 3) + (8 - 2 n) A(n - 4) + (-15 + 3 n) A(n - 5) + (n - 4) A(n - 6) + (7 - n) A(-7 + n) - A(n - 8) In Maple input format: A(n) = (-1+n)*A(-1+n)+(2+n)*A(n-2)+(13-3*n)*A(n-3)+(8-2*n)*A(n-4)+(-15+3*n)*A(n -5)+(n-4)*A(n-6)+(7-n)*A(-7+n)-A(n-8) valid for n>=, 9 and with initial values A(1) = 0, A(2) = 0, A(3) = 0, A(4) = 1, A(5) = 5, A(6) = 33, A(7) = 236, A(8) = 1918 Theorem Number, 12, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-2, -1, 1} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 422368833345419228119321429971945, 13503062209833940507331110098344238, 445205604483564570193623804371379672, 15124321438320496055524588975256112834, 528932713723840211677418119742130987552 , 19027332916172173711808021207782567969321, 703512272753441945059248284824352179963862, 26715484542964602714526573814197955737846696, 1041238004624047356567377284705742284437448391, 41624195980762796383622346494370462920478970264] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-2, -1, 1} is the following rational function: 4 4 3 3 2 2 2 / - (u t - 1) (t u - u t + t u - u t - 2 u t + 1) / (1 - u - 3 u t / 2 2 4 2 4 3 4 4 5 4 5 5 7 6 7 7 + 2 t u + u t + 4 u t + 2 t u - 2 u t - 2 u t - u t - u t 8 8 + t u ) and in Maple input form -(u*t-1)*(t^4*u^4-u^3*t^3+t^2*u^2-u^2*t-2*u*t+1)/(1-u-3*u*t+2*t^2*u^2+u^4*t^2+4 *u^4*t^3+2*t^4*u^4-2*u^5*t^4-2*u^5*t^5-u^7*t^6-u^7*t^7+t^8*u^8) Just for fun the number of such permutations whose length is, 300, equals: 1528841341525559403392909828379020371982102544827921476124577677094469869610769\ 2750974514908984793798758745052575207020122837001164506815561934922262129338291\ 6018890782142423914663120673890108583642169678516096555993029689496781538691042\ 6985442344025080675553311791645290384992100621847171247876918152477908604464051\ 0117104946240618429432592488796334582328755566612118884573240368986350974819531\ 3611772041828528557288079081870582562060267275771015830136135569318706164226390\ 5116851104968924240347916273360378044485818156627124958973648397329535372463402\ 9238517243334216665905252548253852817069350833113939072697886 Theorem Number, 13, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-2, -1, 2} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 437003492655351319523344853889523, 13955190155104596536656917185703479, 459627562160153787048761873402626749, 15598823223130174410016868384203356371, 545020464645975971464010690150282668079 , 19588915751740577877019511103831264812882, 723678862742532009860206239319303662030796, 27459891694947460460227751552524985173719560, 1069462473889997908059884155092760275617038271, 42722627460854225875395987235312049505187053469] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-2, -1, 2} is the following rational function: 11 11 9 9 9 8 8 8 8 7 7 7 6 6 - (u t - 1) (u t - 2 u t - 2 u t + t u - u t - u t - u t 7 5 6 5 5 5 6 4 5 4 4 4 5 3 3 3 + u t + u t + 3 u t + u t + 3 u t - t u + u t + u t 4 2 3 2 2 / 2 2 + u t + u t - 2 u t - 2 u t + 1) / (1 - u + 2 t u - 3 u t / 14 14 16 16 5 2 5 3 6 4 6 5 6 6 - 2 t u + t u + u t + 7 u t - 2 u t - 8 u t - 4 u t 7 4 7 5 8 6 8 7 9 6 9 7 9 8 9 9 - u t - 5 u t + u t + 4 u t - u t - 7 u t - 12 u t - 4 u t 10 8 10 9 10 10 11 8 11 9 11 10 + 2 u t + 8 u t + 4 u t + u t + 5 u t + 6 u t 11 11 12 10 12 11 12 12 13 12 13 13 + 2 u t - 2 u t - 4 u t - 2 u t + 2 u t + 2 u t 15 14 15 15 5 4 5 5 7 6 7 7 8 8 - u t - u t + 12 u t + 4 u t - 6 u t - 2 u t + 2 t u ) and in Maple input form -(u*t-1)*(u^11*t^11-2*u^9*t^9-2*u^9*t^8+t^8*u^8-u^8*t^7-u^7*t^7-u^6*t^6+u^7*t^5 +u^6*t^5+3*u^5*t^5+u^6*t^4+3*u^5*t^4-t^4*u^4+u^5*t^3+u^3*t^3+u^4*t^2+u^3*t^2-2* u^2*t-2*u*t+1)/(1-u+2*t^2*u^2-3*u*t-2*t^14*u^14+t^16*u^16+u^5*t^2+7*u^5*t^3-2*u ^6*t^4-8*u^6*t^5-4*u^6*t^6-u^7*t^4-5*u^7*t^5+u^8*t^6+4*u^8*t^7-u^9*t^6-7*u^9*t^ 7-12*u^9*t^8-4*u^9*t^9+2*u^10*t^8+8*u^10*t^9+4*u^10*t^10+u^11*t^8+5*u^11*t^9+6* u^11*t^10+2*u^11*t^11-2*u^12*t^10-4*u^12*t^11-2*u^12*t^12+2*u^13*t^12+2*u^13*t^ 13-u^15*t^14-u^15*t^15+12*u^5*t^4+4*u^5*t^5-6*u^7*t^6-2*u^7*t^7+2*t^8*u^8) Just for fun the number of such permutations whose length is, 300, equals: 1533971912711876176677600449573291588886269802135599800626401230653641534201624\ 7993999533124264983631450392726660182797502844543177144322571238753425177145580\ 9214467614961898162980244832396873519018836027675355521611745919719639227874743\ 6443831951815564107990315859578761461165511903273470323200122117219834047416154\ 1298110720537115475580921048601362131542946702923253660007316471326207242945080\ 5671023970272855803269126087065062507189404244828076840681387615691225462068513\ 1437305112100330597634454107069772380188725196248362540190343753260447805822784\ 3033658780347401043745020653696895164876362711625399015019377 Theorem Number, 14, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-2, 0, 1} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 409185435626391197054081729616005, 13094331357795257384680470353873438, 432124897042847964508691793112087558, 14692618487036815670362869663816163092, 514253585587645958634763318032734138814 , 18513524141585659703966152800583221131683, 685013860289574145125988168805281037981975, 26030999197640514039538222606265612467876409, 1015226018539808168258478860354902551283450591, 40609672976285349131330892618527539345389160088] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-2, 0, 1} is the following rational function: 4 4 2 2 2 / 2 2 - (u t - 1) (t u - t u - u t - u t + 1) / (1 - u - 3 u t + 2 t u / 4 2 4 3 4 4 5 4 5 5 7 6 7 7 8 8 + u t + 4 u t + 2 t u - 2 u t - 2 u t - u t - u t + t u ) and in Maple input form -(u*t-1)*(t^4*u^4-t^2*u^2-u^2*t-u*t+1)/(1-u-3*u*t+2*t^2*u^2+u^4*t^2+4*u^4*t^3+2 *t^4*u^4-2*u^5*t^4-2*u^5*t^5-u^7*t^6-u^7*t^7+t^8*u^8) Just for fun the number of such permutations whose length is, 300, equals: 1523762191604012263307100977094981781343417384737715653085992908820888116215540\ 4454043945431156554341251754285040713695773771503435757510241350070494764221974\ 8833154652468755390808172945777353720119618400618812128730970938839857932017642\ 3267832409285536025194651071706772134387739653719152541726246811535081572421110\ 8667750449194717657013737099603082814299038701882864598247931901159663297234446\ 3031916885893316716428693787592315742541565603363782059536808854624065806456450\ 7757752699360051307302119988453274544708882008432090405914080329409337503320194\ 1473956230226840136590458977726853365451523780858748934352636 Theorem Number, 15, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-2, 0, 2} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 423333522916010674676789667633150, 13531909357883333485248919656011753, 446097173913583243688229025996541892, 15152771462043986008632596009900291488, 529869094630422791848114836489494128772 , 19059091314074456651892870853801636186113, 704621239274064673309326146937293119412732, 26755320176764677223948256779638135714140361, 1042708887856517601783381590693954960250497756, 41679980795940643503872976393055621597973277348] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-2, 0, 2} is the following rational function: 2 2 2 3 3 3 2 4 4 5 5 1 - t u - u t - u t + u t - u t - t u + u t - ------------------------------------------------------------------- 3 3 2 6 6 4 4 4 3 2 (u t + 1 - 2 u t - u - u t) (u t + 2 t u + u t - u t - 1) and in Maple input form -(1-t^2*u^2-u*t-u^2*t+u^3*t^3-u^3*t^2-t^4*u^4+u^5*t^5)/(u^3*t^3+1-2*u*t-u-u^2*t )/(u^6*t^6+2*t^4*u^4+u^4*t^3-u^2*t-1) Just for fun the number of such permutations whose length is, 300, equals: 1528875604078441432902035279352071532433183032034366100472584029976358879298280\ 5458540170663990478307536682464061491792534586450481153382064171984748113963953\ 4806706487102220245993583795860666572098046542245860462943127109344790834547583\ 5921512389096408310325249679672512149009634434957166013253565185180790331930728\ 0874053717759492681571680708938159281567414441184586884968037288927110873068117\ 6455140391266560708694656522450286192132781066270911343173669087828079676396164\ 7938087860548913953351108214644110537288159986040615767655009319745717504101123\ 4472320379047832184364090664121637440815560650165723198714145 Theorem Number, 16, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-2, 1, 2} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 437003492655351319523344853889523, 13955190155104596536656917185703479, 459627562160153787048761873402626749, 15598823223130174410016868384203356371, 545020464645975971464010690150282668079 , 19588915751740577877019511103831264812882, 723678862742532009860206239319303662030796, 27459891694947460460227751552524985173719560, 1069462473889997908059884155092760275617038271, 42722627460854225875395987235312049505187053469] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-2, 1, 2} is the following rational function: 11 11 9 9 9 8 8 8 8 7 7 7 6 6 - (u t - 1) (u t - 2 u t - 2 u t + u t - u t - u t - u t 7 5 6 5 5 5 6 4 5 4 4 4 5 3 3 3 + u t + u t + 3 u t + u t + 3 u t - t u + u t + u t 4 2 3 2 2 / 2 2 + u t + u t - 2 u t - 2 u t + 1) / (1 - u + 2 t u - 3 u t / 6 6 14 14 16 16 5 2 5 3 5 4 6 4 - 4 u t - 2 t u + t u + u t + 7 u t + 12 u t - 2 u t 6 5 7 4 7 5 7 6 7 7 8 6 8 7 - 8 u t - u t - 5 u t - 6 u t - 2 u t + u t + 4 u t 8 8 9 6 9 7 9 8 9 9 10 8 10 9 + 2 u t - u t - 7 u t - 12 u t - 4 u t + 2 u t + 8 u t 10 10 11 8 11 9 11 10 11 11 12 10 + 4 u t + u t + 5 u t + 6 u t + 2 u t - 2 u t 12 11 12 12 13 12 13 13 15 14 15 15 - 4 u t - 2 u t + 2 u t + 2 u t - u t - u t 5 5 + 4 u t ) and in Maple input form -(u*t-1)*(u^11*t^11-2*u^9*t^9-2*u^9*t^8+u^8*t^8-u^8*t^7-u^7*t^7-u^6*t^6+u^7*t^5 +u^6*t^5+3*u^5*t^5+u^6*t^4+3*u^5*t^4-t^4*u^4+u^5*t^3+u^3*t^3+u^4*t^2+u^3*t^2-2* u^2*t-2*u*t+1)/(1-u+2*t^2*u^2-3*u*t-4*u^6*t^6-2*t^14*u^14+t^16*u^16+u^5*t^2+7*u ^5*t^3+12*u^5*t^4-2*u^6*t^4-8*u^6*t^5-u^7*t^4-5*u^7*t^5-6*u^7*t^6-2*u^7*t^7+u^8 *t^6+4*u^8*t^7+2*u^8*t^8-u^9*t^6-7*u^9*t^7-12*u^9*t^8-4*u^9*t^9+2*u^10*t^8+8*u^ 10*t^9+4*u^10*t^10+u^11*t^8+5*u^11*t^9+6*u^11*t^10+2*u^11*t^11-2*u^12*t^10-4*u^ 12*t^11-2*u^12*t^12+2*u^13*t^12+2*u^13*t^13-u^15*t^14-u^15*t^15+4*u^5*t^5) Just for fun the number of such permutations whose length is, 300, equals: 1533971912711876176677600449573291588886269802135599800626401230653641534201624\ 7993999533124264983631450392726660182797502844543177144322571238753425177145580\ 9214467614961898162980244832396873519018836027675355521611745919719639227874743\ 6443831951815564107990315859578761461165511903273470323200122117219834047416154\ 1298110720537115475580921048601362131542946702923253660007316471326207242945080\ 5671023970272855803269126087065062507189404244828076840681387615691225462068513\ 1437305112100330597634454107069772380188725196248362540190343753260447805822784\ 3033658780347401043745020653696895164876362711625399015019377 Theorem Number, 17, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-1, 0, 1} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 395532367975403585500617187617398, 12671538895479415757151094137608764, 418609107947186158553729777459594142, 14247017829578985769375863952100323913, 499116606951647690158323015933937931464 , 17984173262157571365102643061887284741537, 665972294509356736500776080533293911282908, 25326988274268355336988608984194802575650264, 988492565632482993215879138642849107148411948, 39567769551535581356085272066515650993222290729] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-1, 0, 1} is the following rational function: u t - 1 - ---------------------------- 3 3 2 u t + 1 - 2 u t - u - u t and in Maple input form -(u*t-1)/(u^3*t^3+1-2*u*t-u-u^2*t) Just for fun the number of such permutations whose length is, 300, equals: 1518665940554044997150215861160526885315888084357034861707578334398234086423309\ 6851332182532540718520797294279785622869898505758484116674486827927724263957789\ 7936542871948372718758820589326135694096237087434133345628129959190053868384750\ 5614536752896680302170526001190296894698431305609053838190914646661144396888169\ 2170863289967348770171439031619918655532932904151125043397684809100221677268637\ 0365064896263124317236077886857320552816323082164962459507014692661776870084357\ 3327682085367249380628095315685897811825261366150199675193885851993493026722824\ 2229031116063320245529394344460114826846989532113183413906561 A(n), satisfies the following linear recurrence equations with polynomial (of degree <= one!) coefficients A(n) = n A(n - 1) + 4 A(n - 2) + (9 - 3 n) A(n - 3) + (-4 + n) A(-4 + n) + (-10 + 2 n) A(n - 5) + (7 - n) A(n - 6) - A(-7 + n) In Maple input format: A(n) = n*A(n-1)+4*A(n-2)+(9-3*n)*A(n-3)+(-4+n)*A(-4+n)+(-10+2*n)*A(n-5)+(7-n)*A (n-6)-A(-7+n) valid for n>=, 8 and with initial values A(1) = 0, A(2) = 0, A(3) = 0, A(4) = 1, A(5) = 4, A(6) = 29, A(7) = 206 Theorem Number, 18, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-1, 0, 2} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 409185435626391197054081729616005, 13094331357795257384680470353873438, 432124897042847964508691793112087558, 14692618487036815670362869663816163092, 514253585587645958634763318032734138814 , 18513524141585659703966152800583221131683, 685013860289574145125988168805281037981975, 26030999197640514039538222606265612467876409, 1015226018539808168258478860354902551283450591, 40609672976285349131330892618527539345389160088] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-1, 0, 2} is the following rational function: 4 4 2 2 2 / 2 2 - (u t - 1) (t u - t u - u t - u t + 1) / (1 - u - 3 u t + 2 t u / 4 2 4 3 4 4 5 4 5 5 7 6 7 7 8 8 + u t + 4 u t + 2 t u - 2 u t - 2 u t - u t - u t + u t ) and in Maple input form -(u*t-1)*(t^4*u^4-t^2*u^2-u^2*t-u*t+1)/(1-u-3*u*t+2*t^2*u^2+u^4*t^2+4*u^4*t^3+2 *t^4*u^4-2*u^5*t^4-2*u^5*t^5-u^7*t^6-u^7*t^7+u^8*t^8) Just for fun the number of such permutations whose length is, 300, equals: 1523762191604012263307100977094981781343417384737715653085992908820888116215540\ 4454043945431156554341251754285040713695773771503435757510241350070494764221974\ 8833154652468755390808172945777353720119618400618812128730970938839857932017642\ 3267832409285536025194651071706772134387739653719152541726246811535081572421110\ 8667750449194717657013737099603082814299038701882864598247931901159663297234446\ 3031916885893316716428693787592315742541565603363782059536808854624065806456450\ 7757752699360051307302119988453274544708882008432090405914080329409337503320194\ 1473956230226840136590458977726853365451523780858748934352636 Theorem Number, 19, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-1, 1, 2} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 422368833345419228119321429971945, 13503062209833940507331110098344238, 445205604483564570193623804371379672, 15124321438320496055524588975256112834, 528932713723840211677418119742130987552 , 19027332916172173711808021207782567969321, 703512272753441945059248284824352179963862, 26715484542964602714526573814197955737846696, 1041238004624047356567377284705742284437448391, 41624195980762796383622346494370462920478970264] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-1, 1, 2} is the following rational function: 4 4 3 3 2 2 2 / - (u t - 1) (t u - u t + t u - u t - 2 u t + 1) / (1 - u - 3 u t / 2 2 4 2 4 3 4 4 5 4 5 5 7 6 7 7 + 2 t u + u t + 4 u t + 2 t u - 2 u t - 2 u t - u t - u t 8 8 + u t ) and in Maple input form -(u*t-1)*(t^4*u^4-u^3*t^3+t^2*u^2-u^2*t-2*u*t+1)/(1-u-3*u*t+2*t^2*u^2+u^4*t^2+4 *u^4*t^3+2*t^4*u^4-2*u^5*t^4-2*u^5*t^5-u^7*t^6-u^7*t^7+u^8*t^8) Just for fun the number of such permutations whose length is, 300, equals: 1528841341525559403392909828379020371982102544827921476124577677094469869610769\ 2750974514908984793798758745052575207020122837001164506815561934922262129338291\ 6018890782142423914663120673890108583642169678516096555993029689496781538691042\ 6985442344025080675553311791645290384992100621847171247876918152477908604464051\ 0117104946240618429432592488796334582328755566612118884573240368986350974819531\ 3611772041828528557288079081870582562060267275771015830136135569318706164226390\ 5116851104968924240347916273360378044485818156627124958973648397329535372463402\ 9238517243334216665905252548253852817069350833113939072697886 Theorem Number, 20, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {0, 1, 2} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 408290461702265033886375209600444, 13067496008176147709565436787797750, 431293404936392435759266629619185952, 14666022894638868659882008072210107041, 513376309078215882204904416738197847187 , 18483708874174278938151472221941332456129, 683970727473640876057925304995985455940168, 25993460104389728781062433893888150534647764, 1013837552339715632692424896851506601268719472, 40556928588998185722745225030068193448329700177] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {0, 1, 2} is the following rational function: 3 3 2 1 - 2 u t + u t - u t - ---------------------------------------- 3 3 2 (u t - 1) (u t - u t - 2 u t + 1 - u) and in Maple input form -(1-2*u*t+u^3*t^3-u^2*t)/(u*t-1)/(u^3*t^3-u^2*t-2*u*t+1-u) Just for fun the number of such permutations whose length is, 300, equals: 1523728159974578560804352007539109349566119243749248274203537797374262660830053\ 2064436831429958063604578650914884795003622733498737254873057751131683182325768\ 2079350719657933094703245088497668917797021685461913513431577963660173225053177\ 6824958605876926669900237381412052109620646439827131097160454997768956315317714\ 1242101419322760708876080910952444032415129558785956299837128659915971162226465\ 9577709685200164591844737707980003108499553386583327238127950873326416070806663\ 7467259832027592930245878809132793879178387027080651139641037245406609602674517\ 4651814087517012079766463334498321856063729632553498507083521 A(n), satisfies the following linear recurrence equations with polynomial (of degree <= one!) coefficients A(n) = (-1 + n) A(-1 + n) + (2 + n) A(n - 2) + (13 - 3 n) A(n - 3) + (8 - 2 n) A(n - 4) + (-15 + 3 n) A(n - 5) + (n - 4) A(n - 6) + (7 - n) A(-7 + n) - A(n - 8) In Maple input format: A(n) = (-1+n)*A(-1+n)+(2+n)*A(n-2)+(13-3*n)*A(n-3)+(8-2*n)*A(n-4)+(-15+3*n)*A(n -5)+(n-4)*A(n-6)+(7-n)*A(-7+n)-A(n-8) valid for n>=, 9 and with initial values A(1) = 0, A(2) = 0, A(3) = 0, A(4) = 1, A(5) = 5, A(6) = 33, A(7) = 236, A(8) = 1918 Theorem Number, 21, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-2, -1, 0, 1} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 140247432875402971573369231889848, 4498914657236338655814077833679289, 148803818358909234328665416660809880, 5070160119296016281279107397325207344, 177811824720384961559549668663741701840, 6413299431671382932407789343721528205640, 237713413252320750258868116589029513006297, 9048235327801642886146766055575108824696302, 353439340778476132910710882253767835786723954, 14158730335402414130878374002770357528028837611] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-2, -1, 0, 1} is the following rational function: 4 4 2 / 2 2 3 2 - (u t - 1) (t u - u t - 2 u t + 1) / (1 - 4 u t - u + 4 t u + u t / 4 2 4 3 4 4 5 5 5 4 7 6 8 8 + u t + 4 u t + 2 t u - 4 u t - 3 u t - u t + u t ) and in Maple input form -(u*t-1)*(t^4*u^4-u^2*t-2*u*t+1)/(1-4*u*t-u+4*t^2*u^2+u^3*t^2+u^4*t^2+4*u^4*t^3 +2*t^4*u^4-4*u^5*t^5-3*u^5*t^4-u^7*t^6+u^8*t^8) Just for fun the number of such permutations whose length is, 300, equals: 5568018182677289121456415921956622525993933822207080693388007056493118807464560\ 9603395289550167925818566687436500243888997016175995954664143685895570871075661\ 9873952988865578936171926203363382443297640697552838241043942076220934278481737\ 5296375018126220327576952948842983342150874216803438512645041874318010591958406\ 1096164228752098291957154705575921773918340601318480614513579303756014084316675\ 3266386555491160795823848769729453680126262198026682227402024937408123719222079\ 8841165066494085261440607162090393896139588655619770267614934740455969211182925\ 684716455112653044026788812070810092095342530201372448771578 Theorem Number, 22, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-2, -1, 0, 2} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 145254827461679778643828527255280, 4654009542283974432585752290263815, 153762790161635362821087547636747650, 5233679215603251987037831886806493681, 183367382096172796136653601375197094455, 6607608028086412616777662845480609559347, 244703873957557245049360874668630484456693, 9306718761871009915798122126618574220487454, 363255790810795699418846152274279573975029294, 14541351775829865288097022852589177293263892312] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-2, -1, 0, 2} is the following rational function: 10 10 9 9 8 8 7 7 8 7 7 6 - (u t + 1) (u t - 1) (u t - u t + u t - u t - 2 u t + u t 6 6 6 5 5 5 4 4 5 4 6 4 4 3 - 2 u t - 3 u t + 2 u t - 2 t u + 4 u t + u t - u t 3 3 4 2 3 2 2 2 2 / + 3 u t + u t + 3 u t + t u - 2 u t - 3 u t + 1) / (1 - u / 2 2 3 2 5 4 7 6 8 8 6 5 + 3 t u - 4 u t + 2 u t + 14 u t - 15 u t + 8 u t - 16 u t 6 6 7 7 7 5 7 4 8 7 8 6 9 6 9 7 - 8 u t - 4 u t - 8 u t - u t + 12 u t + u t - u t - 4 u t 9 8 10 8 10 9 10 10 11 8 11 9 - 3 u t + 4 u t + 8 u t + 4 u t + u t + 4 u t 11 10 12 12 12 11 12 10 14 14 14 15 + 4 u t - 4 u t - 4 u t - 2 u t - t u - t u 16 16 3 3 5 2 5 3 6 4 4 4 5 5 + t u + 4 u t + u t + 8 u t - 4 u t - 4 t u + 4 u t ) and in Maple input form -(u*t+1)*(u*t-1)*(u^10*t^10-u^9*t^9+u^8*t^8-u^7*t^7-2*u^8*t^7+u^7*t^6-2*u^6*t^6 -3*u^6*t^5+2*u^5*t^5-2*t^4*u^4+4*u^5*t^4+u^6*t^4-u^4*t^3+3*u^3*t^3+u^4*t^2+3*u^ 3*t^2+t^2*u^2-2*u^2*t-3*u*t+1)/(1-u+3*t^2*u^2-4*u*t+2*u^3*t^2+14*u^5*t^4-15*u^7 *t^6+8*u^8*t^8-16*u^6*t^5-8*u^6*t^6-4*u^7*t^7-8*u^7*t^5-u^7*t^4+12*u^8*t^7+u^8* t^6-u^9*t^6-4*u^9*t^7-3*u^9*t^8+4*u^10*t^8+8*u^10*t^9+4*u^10*t^10+u^11*t^8+4*u^ 11*t^9+4*u^11*t^10-4*u^12*t^12-4*u^12*t^11-2*u^12*t^10-t^14*u^14-t^14*u^15+t^16 *u^16+4*u^3*t^3+u^5*t^2+8*u^5*t^3-4*u^6*t^4-4*t^4*u^4+4*u^5*t^5) Just for fun the number of such permutations whose length is, 300, equals: 5586765714339382608704248725843068388700276704132738580719040175083585865932619\ 0897204265758511808143742694705081358577683482071742323710087196082939523077017\ 5556680732240480913287134016914041430594447703851758865722239627950649253285929\ 6037869967539944105829676277762702537304763247346743360501280426906006373838379\ 3760700656591489663706148791770481563005631456041193693435335022008463125339891\ 4788595049877372266011310983692374141057274530608551537959037904720075783880554\ 0386885587961763145955321944700502184891022188092565229458160943208828155923631\ 398475169369227976651080961555464190713115549351600942530949 Theorem Number, 23, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-2, -1, 1, 2} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 150102336465830547250185089977563, 4804303455418843096354007115309293, 158572809445873769922433070070465274, 5392428307011053439166478798567072552, 188765424273379642809924788302151293426, 6796557871495566255643670701545209670461, 251506675338960931116944680713294832827183, 9558443080505859856548537955396056234671305, 372822039237256582702860852010181058944963653, 14914461572443946396949312892321508869607012658] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-2, -1, 1, 2} is the following rational function: 5 5 3 3 3 2 2 2 2 / - (u t - 1) (u t - u t - u t + t u - u t - 2 u t + 1) / (1 - u / 2 2 3 3 3 2 3 4 4 4 3 5 3 - 3 u t + 2 t u - u t - 4 u t - u t + t u + 3 u t + 2 u t 5 4 5 5 6 5 6 6 7 7 7 5 8 8 + 6 u t + 4 u t - 3 u t - 3 u t + u t - u t - 2 u t 8 7 9 8 9 9 10 10 + 2 u t - u t - u t + u t ) and in Maple input form -(u*t-1)*(u^5*t^5-u^3*t^3-u^3*t^2+t^2*u^2-u^2*t-2*u*t+1)/(1-u-3*u*t+2*t^2*u^2-u ^3*t^3-4*u^3*t^2-u^3*t+t^4*u^4+3*u^4*t^3+2*u^5*t^3+6*u^5*t^4+4*u^5*t^5-3*u^6*t^ 5-3*u^6*t^6+u^7*t^7-u^7*t^5-2*u^8*t^8+2*u^8*t^7-u^9*t^8-u^9*t^9+u^10*t^10) Just for fun the number of such permutations whose length is, 300, equals: 5605450759246290094849537287880792047797817325351500790760945413990949775503611\ 8691615190660503976996094974919828487028620758796367149369945153360575258665548\ 2867456520277990073863162718952354897293934780370221825089362959223190812172443\ 0610584789880712996971153450150064454285637127516394326472976070537079115433191\ 6604384671606894009072954154467847265668139039229657277611859078100064505200523\ 7232831153007799403602645990179060909420121063485103837507388584721411519474966\ 7288832653095400944687087297484461012360602889664388568304520993281867355384364\ 186210915261256386027121931929961548769378148372948016063793 Theorem Number, 24, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-2, 0, 1, 2} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 145254827461679778643828527255280, 4654009542283974432585752290263815, 153762790161635362821087547636747650, 5233679215603251987037831886806493681, 183367382096172796136653601375197094455, 6607608028086412616777662845480609559347, 244703873957557245049360874668630484456693, 9306718761871009915798122126618574220487454, 363255790810795699418846152274279573975029294, 14541351775829865288097022852589177293263892312] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-2, 0, 1, 2} is the following rational function: 10 10 9 9 8 8 8 7 7 7 7 6 - (u t + 1) (u t - 1) (u t - u t + u t - 2 u t - u t + u t 6 6 6 5 5 5 4 4 5 4 6 4 4 3 - 2 u t - 3 u t + 2 u t - 2 t u + 4 u t + u t - u t 3 3 4 2 3 2 2 2 2 / + 3 u t + u t + 3 u t + t u - 2 u t - 3 u t + 1) / (1 - u / 2 2 14 14 14 15 16 16 5 2 6 4 + 3 t u - 4 u t - t u - t u + t u + u t - 4 u t 7 6 7 4 8 6 9 6 9 7 10 8 10 9 - 15 u t - u t + u t - u t - 4 u t + 4 u t + 8 u t 11 8 11 9 11 10 12 12 12 11 12 10 + u t + 4 u t + 4 u t - 4 u t - 4 u t - 2 u t 3 2 5 4 8 8 6 5 6 6 7 7 7 5 + 2 u t + 14 u t + 8 u t - 16 u t - 8 u t - 4 u t - 8 u t 8 7 9 8 10 10 3 3 5 3 4 4 5 5 + 12 u t - 3 u t + 4 u t + 4 u t + 8 u t - 4 t u + 4 u t ) and in Maple input form -(u*t+1)*(u*t-1)*(u^10*t^10-u^9*t^9+u^8*t^8-2*u^8*t^7-u^7*t^7+u^7*t^6-2*u^6*t^6 -3*u^6*t^5+2*u^5*t^5-2*t^4*u^4+4*u^5*t^4+u^6*t^4-u^4*t^3+3*u^3*t^3+u^4*t^2+3*u^ 3*t^2+t^2*u^2-2*u^2*t-3*u*t+1)/(1-u+3*t^2*u^2-4*u*t-t^14*u^14-t^14*u^15+t^16*u^ 16+u^5*t^2-4*u^6*t^4-15*u^7*t^6-u^7*t^4+u^8*t^6-u^9*t^6-4*u^9*t^7+4*u^10*t^8+8* u^10*t^9+u^11*t^8+4*u^11*t^9+4*u^11*t^10-4*u^12*t^12-4*u^12*t^11-2*u^12*t^10+2* u^3*t^2+14*u^5*t^4+8*u^8*t^8-16*u^6*t^5-8*u^6*t^6-4*u^7*t^7-8*u^7*t^5+12*u^8*t^ 7-3*u^9*t^8+4*u^10*t^10+4*u^3*t^3+8*u^5*t^3-4*t^4*u^4+4*u^5*t^5) Just for fun the number of such permutations whose length is, 300, equals: 5586765714339382608704248725843068388700276704132738580719040175083585865932619\ 0897204265758511808143742694705081358577683482071742323710087196082939523077017\ 5556680732240480913287134016914041430594447703851758865722239627950649253285929\ 6037869967539944105829676277762702537304763247346743360501280426906006373838379\ 3760700656591489663706148791770481563005631456041193693435335022008463125339891\ 4788595049877372266011310983692374141057274530608551537959037904720075783880554\ 0386885587961763145955321944700502184891022188092565229458160943208828155923631\ 398475169369227976651080961555464190713115549351600942530949 Theorem Number, 25, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-1, 0, 1, 2} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 140247432875402971573369231889848, 4498914657236338655814077833679289, 148803818358909234328665416660809880, 5070160119296016281279107397325207344, 177811824720384961559549668663741701840, 6413299431671382932407789343721528205640, 237713413252320750258868116589029513006297, 9048235327801642886146766055575108824696302, 353439340778476132910710882253767835786723954, 14158730335402414130878374002770357528028837611] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-1, 0, 1, 2} is the following rational function: 4 4 2 / 2 2 3 2 - (u t - 1) (t u - u t - 2 u t + 1) / (1 - 4 u t - u + 4 t u + u t / 4 2 4 3 4 4 5 5 5 4 7 6 8 8 + u t + 4 u t + 2 t u - 4 u t - 3 u t - u t + u t ) and in Maple input form -(u*t-1)*(t^4*u^4-u^2*t-2*u*t+1)/(1-4*u*t-u+4*t^2*u^2+u^3*t^2+u^4*t^2+4*u^4*t^3 +2*t^4*u^4-4*u^5*t^5-3*u^5*t^4-u^7*t^6+u^8*t^8) Just for fun the number of such permutations whose length is, 300, equals: 5568018182677289121456415921956622525993933822207080693388007056493118807464560\ 9603395289550167925818566687436500243888997016175995954664143685895570871075661\ 9873952988865578936171926203363382443297640697552838241043942076220934278481737\ 5296375018126220327576952948842983342150874216803438512645041874318010591958406\ 1096164228752098291957154705575921773918340601318480614513579303756014084316675\ 3266386555491160795823848769729453680126262198026682227402024937408123719222079\ 8841165066494085261440607162090393896139588655619770267614934740455969211182925\ 684716455112653044026788812070810092095342530201372448771578 Theorem Number, 26, : Let , A(n), be the number of permutations pi of length n such that pi[i]-i is never in the set , {-2, -1, 0, 1, 2} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [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, 47966398139217467057774105014170, 1542645417237634607628635270840168, 51146089460752951540803301799019672, 1746591798401846616343454623340947784, 61381815923650955597148665193547432746, 2218266393702726477767514845791469481265, 82373547373580000762743875088585232328224, 3140893108553310023605863846331657652335392, 122890017890794561080979201273290207699425280, 4930594342732558124044727208386182457167234273] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board made from, {-2, -1, 0, 1, 2} is the following rational function: 5 5 3 2 2 / 2 2 - (u t - 1) (u t - u t - u t - 2 u t + 1) / (1 - 4 u t - u + 4 t u / 3 3 2 3 3 4 4 4 3 5 3 5 4 - u t - 4 u t - 2 u t + 4 t u + 5 u t + 2 u t + 6 u t 5 5 6 6 6 5 7 5 8 8 8 7 9 8 + 2 u t - 4 u t - 5 u t - u t - 2 u t + 2 u t - u t 10 10 + u t ) and in Maple input form -(u*t-1)*(u^5*t^5-u^3*t^2-u^2*t-2*u*t+1)/(1-4*u*t-u+4*t^2*u^2-u^3*t-4*u^3*t^2-2 *u^3*t^3+4*t^4*u^4+5*u^4*t^3+2*u^5*t^3+6*u^5*t^4+2*u^5*t^5-4*u^6*t^6-5*u^6*t^5- u^7*t^5-2*u^8*t^8+2*u^8*t^7-u^9*t^8+u^10*t^10) Just for fun the number of such permutations whose length is, 300, equals: 2034588687713172177000307826672603265919296405605879947382437435105620967375827\ 2878211056714307273514099205469245406180940569673163270877754391687091902005079\ 1059629949815776765899448808223465059281475355866304880553544912072538983675562\ 9892151205017977506370388823135892243826402525518316381692819789462981157202750\ 3102310774869422477097654348841376494030751815686092039933023803245961400749325\ 2769744034291163315125036413892440388128997410373728947978375554876710347091904\ 7591952831091425598934084721066075736475635041226082406702609115738238744222963\ 058544256592833341662238374712936035823653343818947226247873 This ends this book, that took, 11417.242, seconds.