On the enumerating sequences for CIRCULAR permutations that avoid moving in the set S for subsets S of cardinality >=2 that are subsets of, {0, 1, 2, 3, 4} and contain 0 By Shalosh B. Ekhad Theorem No., 1, : Let , A(n), be the number of ways of reseating n diners around a round table in such a way that the number of seats, looking clockwise, that each diner moved, 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, 2, 13, 80, 579, 4738, 43387, 439792, 4890741, 59216642, 775596313, 10927434464, 164806435783, 2649391469058, 45226435601207, 817056406224416, 15574618910994665, 312400218671253762, 6577618644576902053, 145051250421230224304, 3343382818203784146955, 80399425364623070680706, 2013619745874493923699123, 52441212770215183676081296, 1418087454121354412691790045, 39762923867612001445482824194, 1154647923129989496658559750193, 34682040826614983472734095531712, 1076377544439444821254633352940175, 34481075598943956929185850329319426, 1139021316022134503795436380243251567, 38763360887576451083282096894245455168, 1357925683976108354812838248065515591633, 48926368181726746427350357974128938839554, 1811711144161235789501336816905011424974653, 68896667977874338233390779975807570251145232, 2688879692613377250447931017322962684269637331, 107627710512932852479215546777103567971049856642] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board (with the circular convention) made from, {0, 1} is the following rational function: 2 2 2 3 3 -1 + u t - t u - u t + u t - ------------------------------- 2 2 1 - u - 2 u t + t u and in Maple input form -(-1+u*t-t^2*u^2-u^2*t+u^3*t^3)/(1-u-2*u*t+t^2*u^2) Just for fun the number of such permutations whose length is, 400, equals: 8644437924656429400352250148386491276623478726444834989103908127007995838014471\ 0564609800777783894273790852622642248242154051006487776903546208823605189945993\ 4864824361048391160047997294570496205877807089246945538083185153172703115791365\ 8085398109535658509976594863721553476127685866983536217503930212863107543606856\ 5864439499775427060839512695957878158260977820701235390116228733966090506964337\ 5552160115882403386658977577956898321331393926044463343359135466011669182870498\ 3871568640153520470462719016173000709169318131592264441440136674378492958515342\ 6423772622986525584642694127391960214483313878371817671672936024671954972719637\ 2707864019363311519309363316210657017039984022278529002786525241865473246431493\ 7379217478651563292952325121092929780851582697401126971473071574793302453896015\ 613937979328129140662445889635255642482177796191546730022936260391403273024002 A(n), satisfies the following linear recurrence equations with polynomial (of degree <= one!) coefficients A(n) = n A(n - 1) + 2 A(n - 2) + (-n + 4) A(n - 3) - A(n - 4) In Maple input format: A(n) = n*A(n-1)+2*A(n-2)+(-n+4)*A(n-3)-A(n-4) valid for n>=, 6 and with initial values A(1) = 0, A(2) = 0, A(3) = 1, A(4) = 2, A(5) = 13 Theorem No., 2, : Let , A(n), be the number of ways of reseating n diners around a round table in such a way that the number of seats, looking clockwise, that each diner moved, 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, 4, 13, 82, 579, 4740, 43387, 439794, 4890741, 59216644, 775596313, 10927434466, 164806435783, 2649391469060, 45226435601207, 817056406224418, 15574618910994665, 312400218671253764, 6577618644576902053, 145051250421230224306, 3343382818203784146955, 80399425364623070680708, 2013619745874493923699123, 52441212770215183676081298, 1418087454121354412691790045, 39762923867612001445482824196, 1154647923129989496658559750193, 34682040826614983472734095531714, 1076377544439444821254633352940175, 34481075598943956929185850329319428, 1139021316022134503795436380243251567, 38763360887576451083282096894245455170, 1357925683976108354812838248065515591633, 48926368181726746427350357974128938839556, 1811711144161235789501336816905011424974653, 68896667977874338233390779975807570251145234, 2688879692613377250447931017322962684269637331, 107627710512932852479215546777103567971049856644] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board (with the circular convention) made from, {0, 2} is the following rational function: 3 2 3 3 4 4 2 2 2 3 4 3 - (1 - u t + 5 u t + 2 u t + t u - u t - t u + 2 u t - u t 5 4 5 3 5 5 6 5 6 6 / - 7 u t - 2 u t - 5 u t + 2 u t + 3 u t ) / ((u t - 1) / 2 2 (u t + 1) (1 - u - 2 u t + t u )) and in Maple input form -(1-u*t+5*u^3*t^2+2*u^3*t^3+t^4*u^4-u^2*t-t^2*u^2+2*u^3*t-u^4*t^3-7*u^5*t^4-2*u ^5*t^3-5*u^5*t^5+2*u^6*t^5+3*u^6*t^6)/(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, 400, equals: 8644437924656429400352250148386491276623478726444834989103908127007995838014471\ 0564609800777783894273790852622642248242154051006487776903546208823605189945993\ 4864824361048391160047997294570496205877807089246945538083185153172703115791365\ 8085398109535658509976594863721553476127685866983536217503930212863107543606856\ 5864439499775427060839512695957878158260977820701235390116228733966090506964337\ 5552160115882403386658977577956898321331393926044463343359135466011669182870498\ 3871568640153520470462719016173000709169318131592264441440136674378492958515342\ 6423772622986525584642694127391960214483313878371817671672936024671954972719637\ 2707864019363311519309363316210657017039984022278529002786525241865473246431493\ 7379217478651563292952325121092929780851582697401126971473071574793302453896015\ 613937979328129140662445889635255642482177796191546730022936260391403273024004 A(n), satisfies the following linear recurrence equations with polynomial (of degree <= one!) coefficients A(n) = n A(n - 1) + 3 A(n - 2) + (6 - 2 n) A(n - 3) - 3 A(n - 4) + (-6 + n) A(n - 5) + A(-6 + n) In Maple input format: A(n) = n*A(n-1)+3*A(n-2)+(6-2*n)*A(n-3)-3*A(n-4)+(-6+n)*A(n-5)+A(-6+n) valid for n>=, 9 and with initial values A(1) = 0, A(2) = 1, A(3) = 1, A(4) = 4, A(5) = 13, A(6) = 82, A(7) = 579, A(8) = 4740 Theorem No., 3, : Let , A(n), be the number of ways of reseating n diners around a round table in such a way that the number of seats, looking clockwise, that each diner moved, is never in the set, {0, 3} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [0, 0, 2, 2, 13, 80, 579, 4738, 43390, 439792, 4890741, 59216648, 775596313, 10927434464, 164806435822, 2649391469058, 45226435601207, 817056406224656, 15574618910994665, 312400218671253762, 6577618644576903790, 145051250421230224304, 3343382818203784146955, 80399425364623070694920, 2013619745874493923699123, 52441212770215183676081296, 1418087454121354412691920206, 39762923867612001445482824194, 1154647923129989496658559750193, 34682040826614983472734096851088, 1076377544439444821254633352940175, 34481075598943956929185850329319426, 1139021316022134503795436380257923790, 38763360887576451083282096894245455168, 1357925683976108354812838248065515591633, 48926368181726746427350357974129116489480, 1811711144161235789501336816905011424974653, 68896667977874338233390779975807570251145232, 2688879692613377250447931017322962686596426270, 107627710512932852479215546777103567971049856642] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board (with the circular convention) made from, {0, 3} is the following rational function: 3 3 4 4 3 2 3 4 3 5 3 - (-1 + u t + 2 u t - 3 t u + 4 u t + 2 u t - 13 u t + 3 u t 5 4 5 5 6 5 6 6 11 9 11 10 11 11 + 6 u t + u t - 18 u t - 7 u t + 3 u t + 9 u t + 7 u t 6 3 6 4 7 3 7 4 7 5 8 5 8 6 - 2 u t - 10 u t + 3 u t + 21 u t + 54 u t - 3 u t - 15 u t 8 7 9 7 9 8 9 9 10 7 10 8 - 25 u t + 2 u t + 6 u t + 6 u t - 3 u t - 15 u t 10 9 10 10 8 8 4 2 7 6 7 7 - 24 u t - 13 u t - 8 t u - 12 u t + 55 u t + 15 u t 4 / 2 2 6 6 3 3 3 2 - 3 u t) / ((1 - u - 2 u t + t u ) (u t - 2 u t - u t + 1)) / and in Maple input form -(-1+u*t+2*u^3*t^3-3*t^4*u^4+4*u^3*t^2+2*u^3*t-13*u^4*t^3+3*u^5*t^3+6*u^5*t^4+u ^5*t^5-18*u^6*t^5-7*u^6*t^6+3*u^11*t^9+9*u^11*t^10+7*u^11*t^11-2*u^6*t^3-10*u^6 *t^4+3*u^7*t^3+21*u^7*t^4+54*u^7*t^5-3*u^8*t^5-15*u^8*t^6-25*u^8*t^7+2*u^9*t^7+ 6*u^9*t^8+6*u^9*t^9-3*u^10*t^7-15*u^10*t^8-24*u^10*t^9-13*u^10*t^10-8*t^8*u^8-\ 12*u^4*t^2+55*u^7*t^6+15*u^7*t^7-3*u^4*t)/(1-u-2*u*t+t^2*u^2)/(u^6*t^6-2*u^3*t^ 3-u^3*t^2+1) Just for fun the number of such permutations whose length is, 400, equals: 8644437924656429400352250148386491276623478726444834989103908127007995838014471\ 0564609800777783894273790852622642248242154051006487776903546208823605189945993\ 4864824361048391160047997294570496205877807089246945538083185153172703115791365\ 8085398109535658509976594863721553476127685866983536217503930212863107543606856\ 5864439499775427060839512695957878158260977820701235390116228733966090506964337\ 5552160115882403386658977577956898321331393926044463343359135466011669182870498\ 3871568640153520470462719016173000709169318131592264441440136674378492958515342\ 6423772622986525584642694127391960214483313878371817671672936024671954972719637\ 2707864019363311519309363316210657017039984022278529002786525241865473246431493\ 7379217478651563292952325121092929780851582697401126971473071574793302453896015\ 613937979328129140662445889635255642482177796191546730022936260391403273024002 Theorem No., 4, : Let , A(n), be the number of ways of reseating n diners around a round table in such a way that the number of seats, looking clockwise, that each diner moved, is never in the set, {0, 4} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [0, 1, 1, 9, 13, 82, 579, 4752, 43387, 439794, 4890741, 59216968, 775596313, 10927434466, 164806435783, 2649391488016, 45226435601207, 817056406224418, 15574618910994665, 312400218673012936, 6577618644576902053, 145051250421230224306, 3343382818203784146955, 80399425364623307547280, 2013619745874493923699123, 52441212770215183676081298, 1418087454121354412691790045, 39762923867612001489192562056, 1154647923129989496658559750193, 34682040826614983472734095531714, 1076377544439444821254633352940175, 34481075598943956929196447895195664, 1139021316022134503795436380243251567, 38763360887576451083282096894245455170, 1357925683976108354812838248065515591633, 48926368181726746427350361242354563737224, 1811711144161235789501336816905011424974653, 68896667977874338233390779975807570251145234, 2688879692613377250447931017322962684269637331, 107627710512932852479215546778353168845734871696] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board (with the circular convention) made from, {0, 4} is the following rational function: 4 4 3 2 3 4 3 5 3 5 4 - (1 - u t - 3 t u + 2 u t + u t - 11 u t + 40 u t + 26 u t 5 5 6 5 6 6 11 9 11 10 11 11 6 3 + 4 u t - 9 u t + u t + 23 u t + 26 u t + 8 u t - 3 u t 6 4 7 3 7 4 7 5 8 5 8 6 8 7 - 10 u t - u t - 6 u t - 11 u t + 84 u t + 136 u t + 89 u t 9 7 9 8 9 9 10 7 10 8 10 9 - 431 u t - 216 u t - 34 u t + 111 u t + 165 u t + 93 u t 10 10 8 8 4 2 7 6 7 7 4 + 13 u t + 19 t u - 10 u t - 8 u t - 4 u t - 3 u t 18 18 17 18 16 18 15 18 13 7 + 15 t u + 28 t u + 18 t u + 4 t u + 4 u t 13 11 14 10 14 9 14 13 14 12 + 490 u t - 34 u t - 4 u t - 143 u t - 181 u t 14 11 15 15 15 14 15 13 16 16 16 15 - 112 u t - 4 u t - 4 u t - u t + 16 u t + 23 u t 16 14 16 13 17 17 17 16 17 15 + 13 u t + 3 u t - 29 u t - 68 u t - 63 u t 17 14 17 13 5 8 3 8 4 9 3 - 26 u t - 4 u t + 4 u t + 3 u t + 25 u t - 4 u t 9 4 9 5 10 5 10 6 11 8 11 7 - 42 u t - 179 u t + 4 u t + 34 u t + 8 u t + u t 12 10 12 9 12 8 12 7 12 11 - 147 u t - 84 u t - 25 u t - 3 u t - 133 u t 13 10 13 9 13 8 12 12 14 14 + 398 u t + 179 u t + 42 u t - 33 t u - 29 t u 5 2 9 6 13 12 13 13 2 / + 22 u t - 388 u t + 306 u t + 60 u t - u t) / ((u t + 1) / 2 2 2 2 4 4 2 2 2 (u t - 1) (t u + 1) (1 - u - 2 u t + t u ) (t u + 1 - 2 t u - u t) 4 4 2 2 2 (t u + 1 + u t + 2 t u )) and in Maple input form -(1-u*t-3*t^4*u^4+2*u^3*t^2+u^3*t-11*u^4*t^3+40*u^5*t^3+26*u^5*t^4+4*u^5*t^5-9* u^6*t^5+u^6*t^6+23*u^11*t^9+26*u^11*t^10+8*u^11*t^11-3*u^6*t^3-10*u^6*t^4-u^7*t ^3-6*u^7*t^4-11*u^7*t^5+84*u^8*t^5+136*u^8*t^6+89*u^8*t^7-431*u^9*t^7-216*u^9*t ^8-34*u^9*t^9+111*u^10*t^7+165*u^10*t^8+93*u^10*t^9+13*u^10*t^10+19*t^8*u^8-10* u^4*t^2-8*u^7*t^6-4*u^7*t^7-3*u^4*t+15*t^18*u^18+28*t^17*u^18+18*t^16*u^18+4*t^ 15*u^18+4*u^13*t^7+490*u^13*t^11-34*u^14*t^10-4*u^14*t^9-143*u^14*t^13-181*u^14 *t^12-112*u^14*t^11-4*u^15*t^15-4*u^15*t^14-u^15*t^13+16*u^16*t^16+23*u^16*t^15 +13*u^16*t^14+3*u^16*t^13-29*u^17*t^17-68*u^17*t^16-63*u^17*t^15-26*u^17*t^14-4 *u^17*t^13+4*u^5*t+3*u^8*t^3+25*u^8*t^4-4*u^9*t^3-42*u^9*t^4-179*u^9*t^5+4*u^10 *t^5+34*u^10*t^6+8*u^11*t^8+u^11*t^7-147*u^12*t^10-84*u^12*t^9-25*u^12*t^8-3*u^ 12*t^7-133*u^12*t^11+398*u^13*t^10+179*u^13*t^9+42*u^13*t^8-33*t^12*u^12-29*t^ 14*u^14+22*u^5*t^2-388*u^9*t^6+306*u^13*t^12+60*u^13*t^13-u^2*t)/(u*t+1)/(u*t-1 )/(t^2*u^2+1)/(1-u-2*u*t+t^2*u^2)/(t^4*u^4+1-2*t^2*u^2-u^2*t)/(t^4*u^4+1+u^2*t+ 2*t^2*u^2) Just for fun the number of such permutations whose length is, 400, equals: 8644437924656429400352250148386491276623478726444834989103908127007995838014471\ 0564609800777783894273790852622642248242154051006487776903546208823605189945993\ 4864824361048391160047997294570496205877807089246945538083185153172703115791365\ 8085398109535658509976594863721553476127685866983536217503930212863107543606856\ 5864439499775427060839512695957878158260977820701235390116228733966090506964337\ 5552160115882403386658977577956898321331393926044463343359135466011669182870498\ 3871568640153520470887511976424375187219984338420978130842955035973026504734125\ 9703309937230546548630486254767070449170851930873892553350408887026183258374293\ 5966068090585025477315427330076042567026971062191204459035008949298311776402684\ 1768838011062708355512052739134269634915255054546294877382401683816775391183914\ 480089070452038806524144073041433447311027551017830426745037957835308530048016 Theorem No., 5, : Let , A(n), be the number of ways of reseating n diners around a round table in such a way that the number of seats, looking clockwise, that each diner moved, 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, 2, 20, 144, 1265, 12072, 126565, 1445100, 17875140, 238282730, 3407118041, 52034548064, 845569542593, 14570246018686, 265397214435860, 5095853023109484, 102877234050493609, 2178674876680100744, 48296053720501168037 , 1118480911876659396600, 27012357369486579075844, 679192344651429663510262, 17752214070309648660242257, 481640300664961181281424256, 13546525138532752664834181025, 394485386492593863034028504858, 11880374396893285022387630686100, 369620168041784245895021556023784, 11867701438858797521809666206294209, 392869648298809266720099521071616040, 13397102558601680313614481786999560773, 470203170270317138997112729566977311524 , 16971667661830718586462210092362989371076, 629503993367959826314631459105345184302114, 23977030992583231156237861017264718948511465, 937171617718212892390255046935606099990399328, 37565421435130418446037410881796832593260502049] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board (with the circular convention) made from, {0, 1, 2} is the following rational function: 2 2 3 2 3 3 4 4 3 4 3 5 4 - (1 - 2 u t + t u + 7 u t + 3 u t - 5 t u + 2 u t - 6 u t - 5 u t 5 3 5 5 6 5 6 6 / - 2 u t - u t + 2 u t + 3 u t ) / ((u t - 1) / 3 3 2 (u t - u t - 2 u t - u + 1)) and in Maple input form -(1-2*u*t+t^2*u^2+7*u^3*t^2+3*u^3*t^3-5*t^4*u^4+2*u^3*t-6*u^4*t^3-5*u^5*t^4-2*u ^5*t^3-u^5*t^5+2*u^6*t^5+3*u^6*t^6)/(u*t-1)/(u^3*t^3-u^2*t-2*u*t-u+1) Just for fun the number of such permutations whose length is, 400, equals: 3164170721074599768805245387258216819993942560550539955592819468501511300956982\ 5806956151424688339521382155277829685238829915741968191468099505719077184370964\ 0611879174668439590330108954184874679248933106608948848541947597622882663606918\ 4076900655750816042463400753883895341044567181740547326653240450939292038887591\ 9994320759112881645322214608837258231586130269682894839392181485629587377176764\ 1515317747237359386051291405700306650903953459188606063762786110216992528842389\ 7687755097364997299429275805676018318127619190125799213346046647995373483783937\ 1081041457550740221972738138851071515759029303974912556613292958116570027608007\ 6898744227266023528516012657209870423277280529384484900184834062129750269702326\ 7272560910022186914882870534371936514363136925102549954832105967509251492974702\ 981726363576009768061530970492914778150118330308268453723431231561942102920129 A(n), satisfies the following linear recurrence equations with polynomial (of degree <= one!) coefficients A(n) = (n + 1) A(n - 1) + (-n + 11) A(n - 2) + (-7 n + 17) A(n - 3) + (4 n - 39) A(n - 4) + (14 n - 74) A(n - 5) + (-8 n + 66) A(n - 6) + (-10 n + 62) A(n - 7) + (8 n - 81) A(n - 8) + (n + 1) A(n - 9) + (-3 n + 35) A(n - 10) + (n - 15) A(n - 11) + A(n - 12) In Maple input format: A(n) = (n+1)*A(n-1)+(-n+11)*A(n-2)+(-7*n+17)*A(n-3)+(4*n-39)*A(n-4)+(14*n-74)*A (n-5)+(-8*n+66)*A(n-6)+(-10*n+62)*A(n-7)+(8*n-81)*A(n-8)+(n+1)*A(n-9)+(-3*n+35) *A(n-10)+(n-15)*A(n-11)+A(n-12) valid for n>=, 15 and with initial values A(1) = 0, A(2) = 0, A(3) = 0, A(4) = 1, A(5) = 2, A(6) = 20, A(7) = 144, A(8) = 1265, A(9) = 12072, A(10) = 126565, A(11) = 1445100, A(12) = 17875140, A(13) = 238282730, A(14) = 3407118041 Theorem No., 6, : Let , A(n), be the number of ways of reseating n diners around a round table in such a way that the number of seats, looking clockwise, that each diner moved, is never in the set, {0, 1, 3} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [0, 0, 1, 1, 2, 17, 144, 1249, 12006, 126125, 1441888, 17847657, 238019298, 3404316088, 52001848016, 845154325281, 14564549139574, 265313246182181, 5094530031699980, 102855047082584249, 2178280324043183736, 48288638312758659645 , 1118334051990694257716, 27009300638208116324289, 679125639699909648318662, 17750691198601720706675593, 481604000010431373825542604, 13545623306035396933693933256, 394462074687072071543869348906, 11879748361031864591653098871817, 369602726680332159125860244415716, 11867197997961799944134771541093761, 392854610911362146274976171686017246, 13396638303535391989645915916775685337, 470188371075975960285675212886185503776 , 16971181050069449881348393142655713892993, 629487504810932578090587862144478053457194, 23976455745084662365629022764162665404607901, 937150971648032299862068546688412745931026420, 37564659725709024386452823361922185715009246033] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board (with the circular convention) made from, {0, 1, 3} is the following rational function: 3 3 2 2 4 4 3 2 3 4 3 5 3 - (-1 + 2 u t + 2 u t - t u - 9 t u + 3 u t + u t - 27 u t + 8 u t 5 4 5 5 6 5 6 6 8 8 4 2 7 6 + 24 u t + 10 u t + 12 u t + 2 u t - 9 t u - 18 u t + 36 u t 7 7 4 6 3 6 4 7 3 7 4 7 5 + 2 u t - 3 u t + 2 u t + 11 u t + 3 u t + 24 u t + 57 u t 8 5 8 6 8 7 9 7 9 8 9 9 10 7 - 6 u t - 30 u t - 40 u t - 2 u t - 5 u t - u t - 3 u t 10 8 10 9 10 10 11 9 11 10 11 11 - 15 u t - 17 u t - 4 u t + 3 u t + 12 u t + 7 u t ) / 2 2 4 2 4 3 4 4 5 4 / (1 - 3 u t - u + 2 t u + u t + 4 u t + 2 t u - 2 u t / 5 5 7 6 7 7 8 8 - 2 u t - u t - u t + t u ) and in Maple input form -(-1+2*u*t+2*u^3*t^3-t^2*u^2-9*t^4*u^4+3*u^3*t^2+u^3*t-27*u^4*t^3+8*u^5*t^3+24* u^5*t^4+10*u^5*t^5+12*u^6*t^5+2*u^6*t^6-9*t^8*u^8-18*u^4*t^2+36*u^7*t^6+2*u^7*t ^7-3*u^4*t+2*u^6*t^3+11*u^6*t^4+3*u^7*t^3+24*u^7*t^4+57*u^7*t^5-6*u^8*t^5-30*u^ 8*t^6-40*u^8*t^7-2*u^9*t^7-5*u^9*t^8-u^9*t^9-3*u^10*t^7-15*u^10*t^8-17*u^10*t^9 -4*u^10*t^10+3*u^11*t^9+12*u^11*t^10+7*u^11*t^11)/(1-3*u*t-u+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, 400, equals: 3164170670377750417660003118731079633926763774615753571465396025884826837390523\ 4747917729238515419124153146904747093161189919158480740977076334778376345463231\ 1374798641385342125802926919847747774123328830335356403959075271397036926433568\ 4721386835331853256601120781151615402952824549390372388335907305397699846848473\ 4660259963718426281820429318984821786295762919160295275904930720039874357562381\ 9749636477710952355675757627518764777612751080001226259867066491550022967926335\ 7440752066145649367519207115981716897346651716048446356351189886278038857653547\ 2332170913018779785945903221070795613012629154488030589904914532323841055978497\ 4210969108876380141108308728937285578492213875253402440108989226886359000280155\ 2838253201452431580185147992555158878843303338251239144007608985627328256792580\ 077650287557281955857763782583787489410317789945031618261872012851332725428545 Theorem No., 7, : Let , A(n), be the number of ways of reseating n diners around a round table in such a way that the number of seats, looking clockwise, that each diner moved, is never in the set, {0, 1, 4} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [0, 0, 1, 2, 2, 17, 144, 1265, 12006, 126125, 1441888, 17847113, 238013578, 3404271029, 52001439536, 845150145569, 14564502209782, 265312670062085, 5094522340324800, 102854936267366057, 2178278611487531810, 48288610066867303237 , 1118333556927246992576, 27009291451897598286177, 679125459821841559966462, 17750687492494666151241129, 481603919871941768278235340, 13545621491599244587316766209, 394462031762158025447398262026, 11879747301981713426451610037072, 369602699477989693265136996414208, 11867197271715071212533195696335297, 392854590787555935030249266374916102, 13396637725582776767118550736086341661, 470188353893670813062289304913540417884 , 16971180521907822233164317942353492941913, 629487488043294025898974895907562534116730, 23976455195858903506209228328527187819734889, 937150953104774247316611485982554713008939828, 37564659080968621953243660788271787826558176193] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board (with the circular convention) made from, {0, 1, 4} is the following rational function: 3 3 2 2 17 20 20 20 19 20 - (-1 + 2 u t + 2 u t - t u + 4 t u + 15 t u + 44 t u 18 20 18 18 18 17 18 16 18 15 + 26 t u - 34 u t - 105 u t - 67 u t - 11 u t 19 19 19 18 19 17 19 16 19 15 12 9 - 8 u t - 47 u t - 67 u t - 30 u t - 4 u t + 35 u t 12 8 12 7 13 10 13 9 13 8 13 7 + 9 u t + u t - 590 u t - 266 u t - 54 u t - 4 u t 13 11 14 10 14 9 14 13 14 12 - 592 u t + 111 u t + 11 u t + 350 u t + 585 u t 14 11 15 10 15 9 15 13 15 12 + 393 u t + 46 u t + 4 u t + 327 u t + 363 u t 15 11 16 15 16 14 16 13 16 12 + 192 u t - 116 u t - 228 u t - 194 u t - 68 u t 16 11 17 17 17 16 17 15 17 14 - 8 u t + 15 u t + 89 u t + 132 u t + 60 u t 17 13 11 8 12 10 12 11 12 12 + 8 u t - 382 u t + 48 u t + 16 u t - 4 u t 13 12 13 13 15 14 15 15 5 8 3 - 214 u t - 6 u t + 122 u t + 4 u t - 4 u t + 3 u t 8 4 9 3 9 4 9 5 10 5 10 6 + 33 u t + 4 u t + 54 u t + 266 u t - 11 u t - 111 u t 11 7 11 6 11 5 14 14 16 16 5 2 - 197 u t - 46 u t - 4 u t + 46 t u - 9 t u - 34 u t 9 6 3 2 3 4 3 5 3 5 4 5 5 + 582 u t + 3 u t + u t + 5 u t - 88 u t - 75 u t - 15 u t 6 5 6 6 8 8 4 2 7 6 7 7 4 + 72 u t + 18 u t + 26 t u + 5 u t - 8 u t - 8 u t + u t 6 3 6 4 7 3 7 4 7 5 8 5 + 10 u t + 56 u t + 2 u t + 11 u t + 11 u t + 121 u t 8 6 8 7 9 7 9 8 9 9 10 7 + 180 u t + 116 u t + 534 u t + 142 u t - 2 u t - 397 u t 10 8 10 9 10 10 11 9 11 10 / - 584 u t - 298 u t - 40 u t - 321 u t - 94 u t ) / (1 / 2 2 11 8 12 10 12 11 12 12 - u - 3 u t + 2 t u + u t - 2 u t - 4 u t - 2 u t 13 12 13 13 15 14 15 15 14 14 16 16 5 2 + 2 u t + 2 u t - u t - u t - 2 t u + t u + u t 9 6 5 3 5 4 5 5 6 5 6 6 8 8 - u t + 7 u t + 12 u t + 4 u t - 8 u t - 4 u t + 2 t u 7 6 7 7 6 4 7 4 7 5 8 6 8 7 - 6 u t - 2 u t - 2 u t - u t - 5 u t + u t + 4 u t 9 7 9 8 9 9 10 8 10 9 10 10 - 7 u t - 12 u t - 4 u t + 2 u t + 8 u t + 4 u t 11 9 11 10 11 11 + 5 u t + 6 u t + 2 u t ) and in Maple input form -(-1+2*u*t+2*u^3*t^3-t^2*u^2+4*t^17*u^20+15*t^20*u^20+44*t^19*u^20+26*t^18*u^20 -34*u^18*t^18-105*u^18*t^17-67*u^18*t^16-11*u^18*t^15-8*u^19*t^19-47*u^19*t^18-\ 67*u^19*t^17-30*u^19*t^16-4*u^19*t^15+35*u^12*t^9+9*u^12*t^8+u^12*t^7-590*u^13* t^10-266*u^13*t^9-54*u^13*t^8-4*u^13*t^7-592*u^13*t^11+111*u^14*t^10+11*u^14*t^ 9+350*u^14*t^13+585*u^14*t^12+393*u^14*t^11+46*u^15*t^10+4*u^15*t^9+327*u^15*t^ 13+363*u^15*t^12+192*u^15*t^11-116*u^16*t^15-228*u^16*t^14-194*u^16*t^13-68*u^ 16*t^12-8*u^16*t^11+15*u^17*t^17+89*u^17*t^16+132*u^17*t^15+60*u^17*t^14+8*u^17 *t^13-382*u^11*t^8+48*u^12*t^10+16*u^12*t^11-4*u^12*t^12-214*u^13*t^12-6*u^13*t ^13+122*u^15*t^14+4*u^15*t^15-4*u^5*t+3*u^8*t^3+33*u^8*t^4+4*u^9*t^3+54*u^9*t^4 +266*u^9*t^5-11*u^10*t^5-111*u^10*t^6-197*u^11*t^7-46*u^11*t^6-4*u^11*t^5+46*t^ 14*u^14-9*t^16*u^16-34*u^5*t^2+582*u^9*t^6+3*u^3*t^2+u^3*t+5*u^4*t^3-88*u^5*t^3 -75*u^5*t^4-15*u^5*t^5+72*u^6*t^5+18*u^6*t^6+26*t^8*u^8+5*u^4*t^2-8*u^7*t^6-8*u ^7*t^7+u^4*t+10*u^6*t^3+56*u^6*t^4+2*u^7*t^3+11*u^7*t^4+11*u^7*t^5+121*u^8*t^5+ 180*u^8*t^6+116*u^8*t^7+534*u^9*t^7+142*u^9*t^8-2*u^9*t^9-397*u^10*t^7-584*u^10 *t^8-298*u^10*t^9-40*u^10*t^10-321*u^11*t^9-94*u^11*t^10)/(1-u-3*u*t+2*t^2*u^2+ u^11*t^8-2*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^15-2*t^14*u^14+t^16*u^16+u^5*t^2-u^9*t^6+7*u^5*t^3+12*u^5*t^4+4*u^5*t^5-\ 8*u^6*t^5-4*u^6*t^6+2*t^8*u^8-6*u^7*t^6-2*u^7*t^7-2*u^6*t^4-u^7*t^4-5*u^7*t^5+u ^8*t^6+4*u^8*t^7-7*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+5*u^11*t^9+6*u^11*t^10+2*u^11*t^11) Just for fun the number of such permutations whose length is, 400, equals: 3164170670377424648398866678571143419626464459051314009628345836075162598696873\ 5989903039474413716051986047380674719970205884574053688693755895609106927976358\ 8638649946477390549798324521099657574281086165552823764059092463819315600739968\ 9704447453402046547251138153645506931991775036808316300794390258403204389896470\ 2673217367023225595446916506273689386547319513271537172140225882655525993728382\ 5757775370187105671824552380783848502840613484992108679021443852829762006741764\ 9241859185805941925406830536780244836606309057299884029687641264201258615793172\ 0831682172847887330285775890587070913968693555242366402504731152643901324479981\ 6285064745468869839311258524334556966559680916718486198410864407602955135794906\ 8815819459423147745701311308586640754329651164288602897222835347487441283984350\ 372032209577442398598797230141536667264728368646585946747774118840026249747457 Theorem No., 8, : Let , A(n), be the number of ways of reseating n diners around a round table in such a way that the number of seats, looking clockwise, that each diner moved, is never in the set, {0, 2, 3} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [0, 0, 1, 1, 2, 17, 144, 1249, 12006, 126125, 1441888, 17847657, 238019298, 3404316088, 52001848016, 845154325281, 14564549139574, 265313246182181, 5094530031699980, 102855047082584249, 2178280324043183736, 48288638312758659645 , 1118334051990694257716, 27009300638208116324289, 679125639699909648318662, 17750691198601720706675593, 481604000010431373825542604, 13545623306035396933693933256, 394462074687072071543869348906, 11879748361031864591653098871817, 369602726680332159125860244415716, 11867197997961799944134771541093761, 392854610911362146274976171686017246, 13396638303535391989645915916775685337, 470188371075975960285675212886185503776 , 16971181050069449881348393142655713892993, 629487504810932578090587862144478053457194, 23976455745084662365629022764162665404607901, 937150971648032299862068546688412745931026420, 37564659725709024386452823361922185715009246033] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board (with the circular convention) made from, {0, 2, 3} is the following rational function: 3 3 3 2 4 3 5 3 5 4 5 5 - (-1 + 2 u t - u t - u t - 25 u t + 8 u t + 24 u t + 10 u t 6 5 6 6 4 4 2 8 8 4 2 7 6 + 18 u t + 4 u t - 7 u t + u t - 9 t u - 18 u t + 32 u t 4 6 3 6 4 7 3 7 4 7 5 8 5 - 3 u t + 3 u t + 16 u t + 3 u t + 24 u t + 55 u t - 6 u t 8 6 8 7 9 7 9 8 9 9 10 7 10 8 - 30 u t - 40 u t - 3 u t - 7 u t - 2 u t - 3 u t - 15 u t 10 9 10 10 11 9 11 10 11 11 / - 16 u t - 3 u t + 3 u t + 12 u t + 7 u t ) / (1 / 2 2 4 2 4 3 4 4 5 4 5 5 - 3 u t - u + 2 t u + u t + 4 u t + 2 u t - 2 u t - 2 u t 7 6 7 7 8 8 - u t - u t + t u ) and in Maple input form -(-1+2*u*t-u^3*t^3-u^3*t^2-25*u^4*t^3+8*u^5*t^3+24*u^5*t^4+10*u^5*t^5+18*u^6*t^ 5+4*u^6*t^6-7*u^4*t^4+u^2*t-9*t^8*u^8-18*u^4*t^2+32*u^7*t^6-3*u^4*t+3*u^6*t^3+ 16*u^6*t^4+3*u^7*t^3+24*u^7*t^4+55*u^7*t^5-6*u^8*t^5-30*u^8*t^6-40*u^8*t^7-3*u^ 9*t^7-7*u^9*t^8-2*u^9*t^9-3*u^10*t^7-15*u^10*t^8-16*u^10*t^9-3*u^10*t^10+3*u^11 *t^9+12*u^11*t^10+7*u^11*t^11)/(1-3*u*t-u+2*t^2*u^2+u^4*t^2+4*u^4*t^3+2*u^4*t^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, 400, equals: 3164170670377750417660003118731079633926763774615753571465396025884826837390523\ 4747917729238515419124153146904747093161189919158480740977076334778376345463231\ 1374798641385342125802926919847747774123328830335356403959075271397036926433568\ 4721386835331853256601120781151615402952824549390372388335907305397699846848473\ 4660259963718426281820429318984821786295762919160295275904930720039874357562381\ 9749636477710952355675757627518764777612751080001226259867066491550022967926335\ 7440752066145649367519207115981716897346651716048446356351189886278038857653547\ 2332170913018779785945903221070795613012629154488030589904914532323841055978497\ 4210969108876380141108308728937285578492213875253402440108989226886359000280155\ 2838253201452431580185147992555158878843303338251239144007608985627328256792580\ 077650287557281955857763782583787489410317789945031618261872012851332725428545 Theorem No., 9, : Let , A(n), be the number of ways of reseating n diners around a round table in such a way that the number of seats, looking clockwise, that each diner moved, is never in the set, {0, 2, 4} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [0, 1, 0, 4, 2, 36, 144, 1313, 12072, 126729, 1445100, 17875984, 238282730, 3407122693, 52034548064, 845569581921, 14570246018686, 265397214729348, 5095853023109484, 102877234053837825, 2178674876680100744, 48296053720531627569 , 1118480911876659396600, 27012357369487009201680, 679192344651429663510262, 17752214070309653251726821, 481640300664961181281424256, 13546525138532752741940527449, 394485386492593863034028504858, 11880374396893285023329807368996, 369620168041784245895021556023784, 11867701438858797521827988393065857, 392869648298809266720099521071616040, 13397102558601680313614733802331529401, 470203170270317138997112729566977311524 , 16971667661830718586462215659625289854992, 629503993367959826314631459105345184302114, 23977030992583231156237861102363874888126181, 937171617718212892390255046935606099990399328, 37565421435130418446037410883901348138585125681] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board (with the circular convention) made from, {0, 2, 4} is the following rational function: 3 3 2 2 3 2 4 3 5 3 5 4 - (-1 + 2 u t - 8 u t + 2 t u - 10 u t + 24 u t - 84 u t - 61 u t 5 5 6 5 6 6 4 4 2 8 8 4 2 - 5 u t - 22 u t - 24 u t + 8 u t + 2 u t - 67 t u + 16 u t 7 6 7 7 4 6 3 6 4 7 3 7 4 + 307 u t + 85 u t + 3 u t + u t - u t + 15 u t + 128 u t 7 5 8 5 8 6 8 7 9 7 9 8 + 335 u t - 144 u t - 349 u t - 308 u t + 581 u t + 180 u t 9 9 10 7 10 8 10 9 10 10 - 16 u t - 238 u t - 194 u t + 104 u t + 88 u t 11 9 11 10 11 11 20 20 19 20 - 1435 u t - 966 u t - 210 u t + 21 t u + 44 t u 18 20 17 20 16 15 16 14 16 13 + 26 t u + 4 t u - 392 u t - 382 u t - 177 u t 16 12 16 11 17 17 17 16 17 15 - 42 u t - 4 u t + 15 u t + 11 u t - 22 u t 17 14 17 13 18 18 18 17 18 16 18 15 - 22 u t - 4 u t + 6 u t - 2 u t - 5 u t - u t 19 19 19 18 19 17 19 16 19 15 9 3 - 19 u t - 61 u t - 69 u t - 30 u t - 4 u t + 4 u t 9 4 9 5 10 5 10 6 11 7 11 6 + 54 u t + 268 u t - 9 u t - 81 u t - 368 u t - 62 u t 11 5 12 9 12 8 12 7 13 10 13 9 - 4 u t + 542 u t + 109 u t + 7 u t - 277 u t - 140 u t 13 8 13 7 13 11 14 10 14 9 14 14 - 38 u t - 4 u t - 291 u t + 77 u t + 9 u t - 80 u t 14 13 14 12 14 11 15 10 15 9 - 18 u t + 192 u t + 210 u t + 46 u t + 4 u t 15 13 15 12 15 11 5 2 9 6 + 636 u t + 473 u t + 205 u t - 34 u t + 593 u t 11 8 12 10 12 11 12 12 13 13 - 1031 u t + 1057 u t + 840 u t + 171 u t + 16 u t 13 12 15 14 15 15 3 5 8 3 - 138 u t + 466 u t + 148 u t - 3 u t - 4 u t - 3 u t 8 4 16 16 / 3 3 2 - 31 u t - 132 t u ) / ((u t - 1) (u t + 1 - u - 2 u t - u t) / 6 6 2 4 4 4 3 6 6 2 2 2 4 3 (u t - 1 - u t + 2 u t + u t ) (u t + 1 - 2 t u - u t - u t )) and in Maple input form -(-1+2*u*t-8*u^3*t^3+2*t^2*u^2-10*u^3*t^2+24*u^4*t^3-84*u^5*t^3-61*u^5*t^4-5*u^ 5*t^5-22*u^6*t^5-24*u^6*t^6+8*u^4*t^4+2*u^2*t-67*t^8*u^8+16*u^4*t^2+307*u^7*t^6 +85*u^7*t^7+3*u^4*t+u^6*t^3-u^6*t^4+15*u^7*t^3+128*u^7*t^4+335*u^7*t^5-144*u^8* t^5-349*u^8*t^6-308*u^8*t^7+581*u^9*t^7+180*u^9*t^8-16*u^9*t^9-238*u^10*t^7-194 *u^10*t^8+104*u^10*t^9+88*u^10*t^10-1435*u^11*t^9-966*u^11*t^10-210*u^11*t^11+ 21*t^20*u^20+44*t^19*u^20+26*t^18*u^20+4*t^17*u^20-392*u^16*t^15-382*u^16*t^14-\ 177*u^16*t^13-42*u^16*t^12-4*u^16*t^11+15*u^17*t^17+11*u^17*t^16-22*u^17*t^15-\ 22*u^17*t^14-4*u^17*t^13+6*u^18*t^18-2*u^18*t^17-5*u^18*t^16-u^18*t^15-19*u^19* t^19-61*u^19*t^18-69*u^19*t^17-30*u^19*t^16-4*u^19*t^15+4*u^9*t^3+54*u^9*t^4+ 268*u^9*t^5-9*u^10*t^5-81*u^10*t^6-368*u^11*t^7-62*u^11*t^6-4*u^11*t^5+542*u^12 *t^9+109*u^12*t^8+7*u^12*t^7-277*u^13*t^10-140*u^13*t^9-38*u^13*t^8-4*u^13*t^7-\ 291*u^13*t^11+77*u^14*t^10+9*u^14*t^9-80*u^14*t^14-18*u^14*t^13+192*u^14*t^12+ 210*u^14*t^11+46*u^15*t^10+4*u^15*t^9+636*u^15*t^13+473*u^15*t^12+205*u^15*t^11 -34*u^5*t^2+593*u^9*t^6-1031*u^11*t^8+1057*u^12*t^10+840*u^12*t^11+171*u^12*t^ 12+16*u^13*t^13-138*u^13*t^12+466*u^15*t^14+148*u^15*t^15-3*u^3*t-4*u^5*t-3*u^8 *t^3-31*u^8*t^4-132*t^16*u^16)/(u*t-1)/(u^3*t^3+1-u-2*u*t-u^2*t)/(u^6*t^6-1-u^2 *t+2*u^4*t^4+u^4*t^3)/(u^6*t^6+1-2*t^2*u^2-u^2*t-u^4*t^3) Just for fun the number of such permutations whose length is, 400, equals: 3164170721074599768805245387258216819993942560550539955592819468501511300956982\ 5806956151424688339521382155277829685238829915741968191468099505719077184370964\ 0611879174668439590330108954184874679248933106608948848541947597622882663606918\ 4076900655750816042463400753883895341044567181740547326653240450939292038887591\ 9994320759112881645322214608837258231586130269682894839392181485629587377176764\ 1515317747237359386051291405700306650903953459188606063762786110216992528842389\ 7687755097364997300089841892955183721333561928141965062067141105521457012554070\ 1414281104075017235563509611891785689913411054904539422202022740020005453058837\ 5595827397941696261321179062288365066704472846595213460379140587014843326920400\ 7257362718197943630924492528261394516890861502339734095144303424874352423221602\ 144202260171184741950602999395409823741504964089989674913270048142261759724641 Theorem No., 10, : Let , A(n), be the number of ways of reseating n diners around a round table in such a way that the number of seats, looking clockwise, that each diner moved, is never in the set, {0, 3, 4} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [0, 0, 1, 2, 2, 17, 144, 1265, 12006, 126125, 1441888, 17847113, 238013578, 3404271029, 52001439536, 845150145569, 14564502209782, 265312670062085, 5094522340324800, 102854936267366057, 2178278611487531810, 48288610066867303237 , 1118333556927246992576, 27009291451897598286177, 679125459821841559966462, 17750687492494666151241129, 481603919871941768278235340, 13545621491599244587316766209, 394462031762158025447398262026, 11879747301981713426451610037072, 369602699477989693265136996414208, 11867197271715071212533195696335297, 392854590787555935030249266374916102, 13396637725582776767118550736086341661, 470188353893670813062289304913540417884 , 16971180521907822233164317942353492941913, 629487488043294025898974895907562534116730, 23976455195858903506209228328527187819734889, 937150953104774247316611485982554713008939828, 37564659080968621953243660788271787826558176193] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board (with the circular convention) made from, {0, 3, 4} is the following rational function: 3 2 4 3 5 3 5 4 5 5 6 5 - (-1 + 2 u t + 2 u t - 3 u t - 86 u t - 69 u t - 13 u t + 72 u t 6 6 4 4 2 8 8 4 2 7 6 7 7 + 18 u t - u t + u t + 26 t u - u t + 8 u t - 4 u t 6 3 6 4 7 3 7 4 7 5 8 5 + 10 u t + 56 u t + 3 u t + 19 u t + 30 u t + 153 u t 8 6 8 7 9 7 9 8 9 9 10 7 + 217 u t + 128 u t + 493 u t + 114 u t - 8 u t - 418 u t 10 8 10 9 10 10 11 9 11 10 - 604 u t - 304 u t - 40 u t - 325 u t - 100 u t 11 11 20 20 19 20 18 20 17 20 - 2 u t + 15 t u + 44 t u + 26 t u + 4 t u 16 15 16 14 16 13 16 12 16 11 - 110 u t - 220 u t - 192 u t - 68 u t - 8 u t 17 17 17 16 17 15 17 14 17 13 + 12 u t + 81 u t + 129 u t + 60 u t + 8 u t 18 18 18 17 18 16 18 15 19 19 - 34 u t - 108 u t - 71 u t - 12 u t - 7 u t 19 18 19 17 19 16 19 15 9 3 9 4 - 44 u t - 66 u t - 30 u t - 4 u t + 4 u t + 54 u t 9 5 10 5 10 6 11 7 11 6 11 5 + 263 u t - 12 u t - 119 u t - 197 u t - 46 u t - 4 u t 12 9 12 8 13 10 13 9 13 8 13 7 + 3 u t - u t - 570 u t - 263 u t - 54 u t - 4 u t 13 11 14 10 14 9 14 14 14 13 - 551 u t + 119 u t + 12 u t + 46 u t + 356 u t 14 12 14 11 15 10 15 9 15 13 + 604 u t + 413 u t + 46 u t + 4 u t + 313 u t 15 12 15 11 5 2 9 6 11 8 + 353 u t + 190 u t - 34 u t + 562 u t - 383 u t 12 10 12 11 12 12 13 12 15 14 + 11 u t + 4 u t - 4 u t - 186 u t + 116 u t 15 15 3 5 8 3 8 4 16 16 / + 4 u t + u t - 4 u t + 4 u t + 43 u t - 9 t u ) / (1 - u / 2 2 5 3 5 4 5 5 6 5 6 6 - 3 u t + 2 t u + 7 u t + 12 u t + 4 u t - 8 u t - 4 u t 8 8 7 6 7 7 6 4 7 4 7 5 8 6 + 2 t u - 6 u t - 2 u t - 2 u t - u t - 5 u t + u t 8 7 9 7 9 8 9 9 10 8 10 9 10 10 + 4 u t - 7 u t - 12 u t - 4 u t + 2 u t + 8 u t + 4 u t 11 9 11 10 11 11 14 14 5 2 9 6 11 8 + 5 u t + 6 u t + 2 u t - 2 u t + u t - u t + u t 12 10 12 11 12 12 13 13 13 12 15 14 - 2 u t - 4 u t - 2 u t + 2 u t + 2 u t - u t 15 15 16 16 - u t + t u ) and in Maple input form -(-1+2*u*t+2*u^3*t^2-3*u^4*t^3-86*u^5*t^3-69*u^5*t^4-13*u^5*t^5+72*u^6*t^5+18*u ^6*t^6-u^4*t^4+u^2*t+26*t^8*u^8-u^4*t^2+8*u^7*t^6-4*u^7*t^7+10*u^6*t^3+56*u^6*t ^4+3*u^7*t^3+19*u^7*t^4+30*u^7*t^5+153*u^8*t^5+217*u^8*t^6+128*u^8*t^7+493*u^9* t^7+114*u^9*t^8-8*u^9*t^9-418*u^10*t^7-604*u^10*t^8-304*u^10*t^9-40*u^10*t^10-\ 325*u^11*t^9-100*u^11*t^10-2*u^11*t^11+15*t^20*u^20+44*t^19*u^20+26*t^18*u^20+4 *t^17*u^20-110*u^16*t^15-220*u^16*t^14-192*u^16*t^13-68*u^16*t^12-8*u^16*t^11+ 12*u^17*t^17+81*u^17*t^16+129*u^17*t^15+60*u^17*t^14+8*u^17*t^13-34*u^18*t^18-\ 108*u^18*t^17-71*u^18*t^16-12*u^18*t^15-7*u^19*t^19-44*u^19*t^18-66*u^19*t^17-\ 30*u^19*t^16-4*u^19*t^15+4*u^9*t^3+54*u^9*t^4+263*u^9*t^5-12*u^10*t^5-119*u^10* t^6-197*u^11*t^7-46*u^11*t^6-4*u^11*t^5+3*u^12*t^9-u^12*t^8-570*u^13*t^10-263*u ^13*t^9-54*u^13*t^8-4*u^13*t^7-551*u^13*t^11+119*u^14*t^10+12*u^14*t^9+46*u^14* t^14+356*u^14*t^13+604*u^14*t^12+413*u^14*t^11+46*u^15*t^10+4*u^15*t^9+313*u^15 *t^13+353*u^15*t^12+190*u^15*t^11-34*u^5*t^2+562*u^9*t^6-383*u^11*t^8+11*u^12*t ^10+4*u^12*t^11-4*u^12*t^12-186*u^13*t^12+116*u^15*t^14+4*u^15*t^15+u^3*t-4*u^5 *t+4*u^8*t^3+43*u^8*t^4-9*t^16*u^16)/(1-u-3*u*t+2*t^2*u^2+7*u^5*t^3+12*u^5*t^4+ 4*u^5*t^5-8*u^6*t^5-4*u^6*t^6+2*t^8*u^8-6*u^7*t^6-2*u^7*t^7-2*u^6*t^4-u^7*t^4-5 *u^7*t^5+u^8*t^6+4*u^8*t^7-7*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+5*u^11*t^9+6*u^11*t^10+2*u^11*t^11-2*u^14*t^14+u^5*t^2-u^9*t^6+u^ 11*t^8-2*u^12*t^10-4*u^12*t^11-2*u^12*t^12+2*u^13*t^13+2*u^13*t^12-u^15*t^14-u^ 15*t^15+t^16*u^16) Just for fun the number of such permutations whose length is, 400, equals: 3164170670377424648398866678571143419626464459051314009628345836075162598696873\ 5989903039474413716051986047380674719970205884574053688693755895609106927976358\ 8638649946477390549798324521099657574281086165552823764059092463819315600739968\ 9704447453402046547251138153645506931991775036808316300794390258403204389896470\ 2673217367023225595446916506273689386547319513271537172140225882655525993728382\ 5757775370187105671824552380783848502840613484992108679021443852829762006741764\ 9241859185805941925406830536780244836606309057299884029687641264201258615793172\ 0831682172847887330285775890587070913968693555242366402504731152643901324479981\ 6285064745468869839311258524334556966559680916718486198410864407602955135794906\ 8815819459423147745701311308586640754329651164288602897222835347487441283984350\ 372032209577442398598797230141536667264728368646585946747774118840026249747457 Theorem No., 11, : Let , A(n), be the number of ways of reseating n diners around a round table in such a way that the number of seats, looking clockwise, that each diner moved, is never in the set, {0, 1, 2, 3} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [0, 0, 0, 0, 1, 2, 31, 264, 2783, 30818, 369321, 4745952, 65275999, 957874226, 14951584189, 247524019720, 4334022049377, 80052395326514, 1555999253409203, 31755107852542144, 679008663143893773, 15182701602959054546, 354364531995856105099, 8618865446674052425224, 218107566239993684272531, 5734302907005373513322674, 156419366922989437119483053, 4421296327690290833867429280, 129342793572697058001361979515, 3911872744020239697091063448610, 122186361171383713228524253528505, 3937574492398733597775756214634504, 130797671713809401672661431633725701, 4474620445439502062225784325579426082, 157520825370053795921028965470658524071, 5701710972929337471229726555336743113088, 212048638027981291325131546552683175743897, 8096977188862533046508859949651226179533410, 317231862755043395438188803396603795429015031, 12744435983915127294453604684178447501994406664] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board (with the circular convention) made from, {0, 1, 2, 3} is the following rational function: 3 3 2 2 3 2 4 3 5 3 5 4 - (-1 + 3 u t - 2 u t - 2 t u - 5 u t - 39 u t + 19 u t + 66 u t 5 5 6 5 4 4 2 8 8 4 2 7 6 + 30 u t + 33 u t - 10 u t + u t - 33 t u - 24 u t + 30 u t 7 7 4 6 3 6 4 7 3 7 4 7 5 - 4 u t - 3 u t + 7 u t + 37 u t + 3 u t + 27 u t + 64 u t 8 5 8 6 8 7 9 7 9 8 9 9 10 7 - 9 u t - 57 u t - 93 u t - 4 u t - 6 u t + 3 u t - 3 u t 10 8 10 9 10 10 11 9 11 10 11 11 - 15 u t - 6 u t + 6 u t + 3 u t + 15 u t + 10 u t 3 / 2 2 3 2 4 2 4 3 4 4 - u t) / (1 - 4 u t - u + 4 t u + u t + u t + 4 u t + 2 u t / 5 5 5 4 7 6 8 8 - 4 u t - 3 u t - u t + t u ) and in Maple input form -(-1+3*u*t-2*u^3*t^3-2*t^2*u^2-5*u^3*t^2-39*u^4*t^3+19*u^5*t^3+66*u^5*t^4+30*u^ 5*t^5+33*u^6*t^5-10*u^4*t^4+u^2*t-33*t^8*u^8-24*u^4*t^2+30*u^7*t^6-4*u^7*t^7-3* u^4*t+7*u^6*t^3+37*u^6*t^4+3*u^7*t^3+27*u^7*t^4+64*u^7*t^5-9*u^8*t^5-57*u^8*t^6 -93*u^8*t^7-4*u^9*t^7-6*u^9*t^8+3*u^9*t^9-3*u^10*t^7-15*u^10*t^8-6*u^10*t^9+6*u ^10*t^10+3*u^11*t^9+15*u^11*t^10+10*u^11*t^11-u^3*t)/(1-4*u*t-u+4*t^2*u^2+u^3*t ^2+u^4*t^2+4*u^4*t^3+2*u^4*t^4-4*u^5*t^5-3*u^5*t^4-u^7*t^6+t^8*u^8) Just for fun the number of such permutations whose length is, 400, equals: 1155270369680731589824799502938739295482870096442741687733820170018631548402051\ 6413569234875058293327763630687823773864364663554124054000793023936424479567579\ 9476587822002722725435981682285762935342518097819470352398696991215682852079111\ 2462896273967912875635766978361483126894231951611438338121649425467083918169001\ 3952847825297108642070616559752521970428397446661611173204715143677604695214659\ 0627980175265859058733006597966329317616133741383120756239920312925866806771435\ 5665407008953631262637649808108776124217106098806047612171374387462225529504394\ 6068624129499065685176688599850902868311847325146578486927866632857607136478302\ 6576077442112235257395469990054797233718591017771927639427823486777590810537303\ 3534618959853228230163732674412353784869187721824923772408387637321115657242984\ 441439196886922338670465138551106992876941670095976037371504720894318730202632 Theorem No., 12, : Let , A(n), be the number of ways of reseating n diners around a round table in such a way that the number of seats, looking clockwise, that each diner moved, is never in the set, {0, 1, 2, 4} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [0, 0, 0, 1, 1, 4, 24, 258, 2693, 30420, 366263, 4721890, 65052009, 955548912, 14924846419, 247188243522, 4329454707079, 79985549463508, 1554952259395455, 31737638114740994, 678699357267709272, 15176910657776895972, 354250232922107379987, 8616493618489525895394, 218055947629175588509131, 5733127334643201215731044, 156391406904279706882366619, 4420603106888536838875212880, 129324907334509407029039846739, 3911393225734166248708583124372, 122173022673435830836148477178305, 3937190036283823056238011547785218, 130786203706497375006183615530229239, 4474266830778691763689997443107122196, 157509566201473642198882091562701036336, 5701341163071319501511405331124460088514, 212036119997026430954729162366834150622279, 8096540881343491809728180651482911673248356, 317216217428868257560714338546265114691752873, 12743859259486651033006885706396735613399084418] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board (with the circular convention) made from, {0, 1, 2, 4} is the following rational function: 3 3 2 2 3 2 4 3 5 3 5 4 - (-1 + 3 u t - 5 u t - t u - 5 u t + 16 u t - 155 u t - 164 u t 5 5 6 5 17 20 20 20 19 20 18 20 - 34 u t + 230 u t + 4 t u + 25 t u + 68 t u + 34 t u 17 16 17 15 18 18 18 17 18 16 - 23 u t - 7 u t - 17 u t - 74 u t - 49 u t 18 15 19 19 19 18 19 17 19 16 - 7 u t + 10 u t - 10 u t - 65 u t - 34 u t 19 15 11 7 11 6 11 5 12 9 12 8 - 4 u t - 401 u t - 66 u t - 4 u t + 448 u t + 55 u t 12 7 13 10 13 9 13 8 13 7 13 13 + u t - 291 u t - 200 u t - 50 u t - 4 u t + 28 u t 13 12 13 11 14 10 14 9 14 13 + 85 u t - 90 u t + 195 u t + 19 u t + 464 u t 14 12 14 11 15 10 15 9 15 15 + 848 u t + 634 u t + 50 u t + 4 u t - 43 u t 15 13 15 12 15 11 16 15 16 14 + 280 u t + 391 u t + 214 u t - 379 u t - 458 u t 16 13 16 12 16 11 17 17 9 6 - 288 u t - 84 u t - 8 u t - 7 u t + 905 u t 11 8 12 10 12 11 12 12 5 - 1058 u t + 1082 u t + 862 u t + 188 u t - 4 u t 8 3 8 4 9 3 9 4 9 5 10 5 + 3 u t + 39 u t + 4 u t + 66 u t + 380 u t - 19 u t 10 6 14 14 14 15 16 16 5 2 6 6 - 239 u t + 55 t u - 14 t u - 93 t u - 46 u t + 55 u t 4 4 2 8 8 4 2 7 6 7 7 4 + 9 u t + u t - 80 t u + 7 u t + 252 u t + 44 u t + u t 6 3 6 4 7 3 7 4 7 5 8 5 + 21 u t + 148 u t + 12 u t + 118 u t + 320 u t + 137 u t 8 6 8 7 9 7 9 8 9 9 10 7 + 53 u t - 193 u t + 831 u t + 218 u t + 12 u t - 1022 u t 10 8 10 9 10 10 11 9 11 10 - 1697 u t - 963 u t - 140 u t - 1134 u t - 367 u t 11 11 3 / 3 3 2 2 3 2 - 8 u t - u t) / (1 - u - 4 u t + 4 u t + 3 t u + 2 u t / 5 3 5 4 5 5 6 5 9 6 11 8 12 10 + 8 u t + 14 u t + 4 u t - 16 u t - u t + u t - 2 u t 12 11 12 12 14 14 14 15 16 16 5 2 6 6 - 4 u t - 4 u t - t u - t u + t u + u t - 8 u t 4 4 8 8 7 6 7 7 6 4 7 4 7 5 - 4 u t + 8 t u - 15 u t - 4 u t - 4 u t - u t - 8 u t 8 6 8 7 9 7 9 8 10 8 10 9 10 10 + u t + 12 u t - 4 u t - 3 u t + 4 u t + 8 u t + 4 u t 11 9 11 10 + 4 u t + 4 u t ) and in Maple input form -(-1+3*u*t-5*u^3*t^3-t^2*u^2-5*u^3*t^2+16*u^4*t^3-155*u^5*t^3-164*u^5*t^4-34*u^ 5*t^5+230*u^6*t^5+4*t^17*u^20+25*t^20*u^20+68*t^19*u^20+34*t^18*u^20-23*u^17*t^ 16-7*u^17*t^15-17*u^18*t^18-74*u^18*t^17-49*u^18*t^16-7*u^18*t^15+10*u^19*t^19-\ 10*u^19*t^18-65*u^19*t^17-34*u^19*t^16-4*u^19*t^15-401*u^11*t^7-66*u^11*t^6-4*u ^11*t^5+448*u^12*t^9+55*u^12*t^8+u^12*t^7-291*u^13*t^10-200*u^13*t^9-50*u^13*t^ 8-4*u^13*t^7+28*u^13*t^13+85*u^13*t^12-90*u^13*t^11+195*u^14*t^10+19*u^14*t^9+ 464*u^14*t^13+848*u^14*t^12+634*u^14*t^11+50*u^15*t^10+4*u^15*t^9-43*u^15*t^15+ 280*u^15*t^13+391*u^15*t^12+214*u^15*t^11-379*u^16*t^15-458*u^16*t^14-288*u^16* t^13-84*u^16*t^12-8*u^16*t^11-7*u^17*t^17+905*u^9*t^6-1058*u^11*t^8+1082*u^12*t ^10+862*u^12*t^11+188*u^12*t^12-4*u^5*t+3*u^8*t^3+39*u^8*t^4+4*u^9*t^3+66*u^9*t ^4+380*u^9*t^5-19*u^10*t^5-239*u^10*t^6+55*t^14*u^14-14*t^14*u^15-93*t^16*u^16-\ 46*u^5*t^2+55*u^6*t^6+9*u^4*t^4+u^2*t-80*t^8*u^8+7*u^4*t^2+252*u^7*t^6+44*u^7*t ^7+u^4*t+21*u^6*t^3+148*u^6*t^4+12*u^7*t^3+118*u^7*t^4+320*u^7*t^5+137*u^8*t^5+ 53*u^8*t^6-193*u^8*t^7+831*u^9*t^7+218*u^9*t^8+12*u^9*t^9-1022*u^10*t^7-1697*u^ 10*t^8-963*u^10*t^9-140*u^10*t^10-1134*u^11*t^9-367*u^11*t^10-8*u^11*t^11-u^3*t )/(1-u-4*u*t+4*u^3*t^3+3*t^2*u^2+2*u^3*t^2+8*u^5*t^3+14*u^5*t^4+4*u^5*t^5-16*u^ 6*t^5-u^9*t^6+u^11*t^8-2*u^12*t^10-4*u^12*t^11-4*u^12*t^12-t^14*u^14-t^14*u^15+ t^16*u^16+u^5*t^2-8*u^6*t^6-4*u^4*t^4+8*t^8*u^8-15*u^7*t^6-4*u^7*t^7-4*u^6*t^4- u^7*t^4-8*u^7*t^5+u^8*t^6+12*u^8*t^7-4*u^9*t^7-3*u^9*t^8+4*u^10*t^8+8*u^10*t^9+ 4*u^10*t^10+4*u^11*t^9+4*u^11*t^10) Just for fun the number of such permutations whose length is, 400, equals: 1155270332285520666800398196417460715079188127765890632639372060334821188014187\ 4174514647262741720403585706987736753591014622128837015674842303804643105306647\ 2765193256242818210051783634768722927760217927168405201638837117717573233861921\ 6800466537788953334354308465687664669025423536573661289549489274677885134659227\ 8958386302512321749821912318288161486349424067241772576650312479042944435366696\ 4716000895620378639809886567551031006323066756728282061700861249832389122511843\ 5084399611436582543097301163901967882039300097891786190274715655448046113786751\ 6940094952192697242391818412052107017886913836919343003129697811134138454621990\ 0545423087611587227266771694979347045630762590302563671439853353890939844076566\ 1462229620739177127272438912969741284977927483628053047233438065634521543155478\ 845690691391999921358022271859973039689860078154440094645447790769311235219714 Theorem No., 13, : Let , A(n), be the number of ways of reseating n diners around a round table in such a way that the number of seats, looking clockwise, that each diner moved, is never in the set, {0, 1, 3, 4} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [0, 0, 1, 1, 1, 8, 31, 260, 2702, 30418, 366131, 4720560, 65043611, 955484090, 14924278834, 247182694788, 4329394211559, 79984819089224, 1554942631156209, 31737500775494756, 678697252118903190, 15176876176577714970, 354249632193536392893, 8616482530211962521488, 218055731525458451531681, 5733122900856085737417202, 156391311394160157099066626, 4420600951856851921217391460, 129324856511948692287090885877, 3911391975437499555788343538120, 122172990643767701045921427087979, 3937189183246107615700331798203652, 130786180122880330310998339307921630, 4474266154884679351772445367177353922, 157509546146762166890512041343759048751, 5701340547743263795921281861555508287792, 212036100495551823321154286395983619798143, 8096540243596892636787505184893917022551114, 317216195929443019259416257136393030178255346, 12743858513025153140951164953151716169421389060] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board (with the circular convention) made from, {0, 1, 3, 4} is the following rational function: 3 3 2 2 3 2 4 3 5 3 5 4 - (-1 + 3 u t + 2 u t - 2 t u + 5 u t - 26 u t - 145 u t - 131 u t 5 5 6 5 17 20 20 20 19 20 18 20 - 23 u t + 248 u t + 4 t u + 27 t u + 68 t u + 34 t u 17 16 17 15 18 18 18 17 18 16 - 6 u t + 244 u t - 102 u t - 282 u t - 163 u t 18 15 19 19 19 17 19 16 19 15 - 22 u t + 14 u t - 62 u t - 34 u t - 4 u t 11 7 11 6 11 5 12 9 12 8 12 7 - 220 u t - 50 u t - 4 u t - 305 u t - 75 u t - 6 u t 13 10 13 9 13 8 13 7 13 13 - 898 u t - 378 u t - 66 u t - 4 u t + 57 u t 13 12 13 11 14 10 14 9 14 13 - 69 u t - 795 u t + 217 u t + 18 u t + 1288 u t 14 12 14 11 15 10 15 9 15 15 + 1707 u t + 930 u t + 50 u t + 4 u t - 26 u t 15 13 15 12 15 11 16 15 16 14 + 240 u t + 330 u t + 202 u t - 139 u t - 279 u t 16 13 16 12 16 11 17 17 9 6 - 320 u t - 118 u t - 12 u t - 53 u t + 750 u t 11 8 12 10 12 11 12 12 5 8 3 - 429 u t - 480 u t - 206 u t + 5 u t - 4 u t + 10 u t 8 4 9 3 9 4 9 5 10 5 10 6 + 125 u t + 4 u t + 66 u t + 354 u t - 26 u t - 285 u t 14 14 14 15 16 16 5 2 6 6 4 4 + 266 t u + 35 t u - 36 t u - 46 u t + 66 u t - 8 u t 2 8 8 4 2 7 6 7 7 4 6 3 + u t + 13 t u - 14 u t + 67 u t - 2 u t - 2 u t + 23 u t 6 4 7 3 7 4 7 5 8 5 8 6 + 157 u t + 10 u t + 83 u t + 173 u t + 479 u t + 643 u t 8 7 9 7 9 8 9 9 10 7 10 8 + 271 u t + 630 u t + 182 u t + 16 u t - 1075 u t - 1765 u t 10 9 10 10 11 9 11 10 11 11 - 1191 u t - 228 u t - 371 u t - 99 u t + 12 u t 17 14 17 13 3 / 2 2 + 136 u t + 16 u t + 2 u t) / ((t u + 1) / 4 4 2 2 2 3 3 10 10 9 9 9 8 (u t + 1 - u t - 2 t u - u t + u t ) (u t - u t - u t 8 8 8 7 7 7 6 6 7 5 6 5 5 5 - 2 t u + 2 u t + u t - 3 u t - u t - 3 u t + 4 u t 5 4 4 4 5 3 4 3 3 3 3 2 2 2 3 + 6 u t + u t + 2 u t + 3 u t - u t - 4 u t + 2 t u - u t - 3 u t + 1 - u)) and in Maple input form -(-1+3*u*t+2*u^3*t^3-2*t^2*u^2+5*u^3*t^2-26*u^4*t^3-145*u^5*t^3-131*u^5*t^4-23* u^5*t^5+248*u^6*t^5+4*t^17*u^20+27*t^20*u^20+68*t^19*u^20+34*t^18*u^20-6*u^17*t ^16+244*u^17*t^15-102*u^18*t^18-282*u^18*t^17-163*u^18*t^16-22*u^18*t^15+14*u^ 19*t^19-62*u^19*t^17-34*u^19*t^16-4*u^19*t^15-220*u^11*t^7-50*u^11*t^6-4*u^11*t ^5-305*u^12*t^9-75*u^12*t^8-6*u^12*t^7-898*u^13*t^10-378*u^13*t^9-66*u^13*t^8-4 *u^13*t^7+57*u^13*t^13-69*u^13*t^12-795*u^13*t^11+217*u^14*t^10+18*u^14*t^9+ 1288*u^14*t^13+1707*u^14*t^12+930*u^14*t^11+50*u^15*t^10+4*u^15*t^9-26*u^15*t^ 15+240*u^15*t^13+330*u^15*t^12+202*u^15*t^11-139*u^16*t^15-279*u^16*t^14-320*u^ 16*t^13-118*u^16*t^12-12*u^16*t^11-53*u^17*t^17+750*u^9*t^6-429*u^11*t^8-480*u^ 12*t^10-206*u^12*t^11+5*u^12*t^12-4*u^5*t+10*u^8*t^3+125*u^8*t^4+4*u^9*t^3+66*u ^9*t^4+354*u^9*t^5-26*u^10*t^5-285*u^10*t^6+266*t^14*u^14+35*t^14*u^15-36*t^16* u^16-46*u^5*t^2+66*u^6*t^6-8*u^4*t^4+u^2*t+13*t^8*u^8-14*u^4*t^2+67*u^7*t^6-2*u ^7*t^7-2*u^4*t+23*u^6*t^3+157*u^6*t^4+10*u^7*t^3+83*u^7*t^4+173*u^7*t^5+479*u^8 *t^5+643*u^8*t^6+271*u^8*t^7+630*u^9*t^7+182*u^9*t^8+16*u^9*t^9-1075*u^10*t^7-\ 1765*u^10*t^8-1191*u^10*t^9-228*u^10*t^10-371*u^11*t^9-99*u^11*t^10+12*u^11*t^ 11+136*u^17*t^14+16*u^17*t^13+2*u^3*t)/(t^2*u^2+1)/(u^4*t^4+1-u*t-2*t^2*u^2-u^2 *t+u^3*t^3)/(u^10*t^10-u^9*t^9-u^9*t^8-2*t^8*u^8+2*u^8*t^7+u^7*t^7-3*u^6*t^6-u^ 7*t^5-3*u^6*t^5+4*u^5*t^5+6*u^5*t^4+u^4*t^4+2*u^5*t^3+3*u^4*t^3-u^3*t^3-4*u^3*t ^2+2*t^2*u^2-u^3*t-3*u*t+1-u) Just for fun the number of such permutations whose length is, 400, equals: 1155270332285159614852688573998623114412469436249900997040433492152343622874610\ 2852794139095753690897932233987494899641101536852292797030678522897573504464803\ 8825144762485381712340452980687920343946127314282318127314033147170801097960482\ 2894784305462731502565425295415678783725785014616072295592427017251689560552956\ 6010412099183711283327904906709562670564869690882418549778732785378137321917490\ 4169548017663995286359258767407696857143275292371073174483960284687447627900364\ 0406482001362453023226748624225659052690553905120450846842352575715464032177400\ 5655107835411548768020810541519209811961575160820378897385226860534101604347323\ 0669949794213136366665019660064460974634380434517398325752068864337455237469357\ 6059190172560057619622751113313438448901841676189015068559810259407416453057775\ 089778738825271944907595833658377536225926957651774318759604496078129182094724 Theorem No., 14, : Let , A(n), be the number of ways of reseating n diners around a round table in such a way that the number of seats, looking clockwise, that each diner moved, is never in the set, {0, 2, 3, 4} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [0, 0, 0, 1, 1, 4, 24, 258, 2693, 30420, 366263, 4721890, 65052009, 955548912, 14924846419, 247188243522, 4329454707079, 79985549463508, 1554952259395455, 31737638114740994, 678699357267709272, 15176910657776895972, 354250232922107379987, 8616493618489525895394, 218055947629175588509131, 5733127334643201215731044, 156391406904279706882366619, 4420603106888536838875212880, 129324907334509407029039846739, 3911393225734166248708583124372, 122173022673435830836148477178305, 3937190036283823056238011547785218, 130786203706497375006183615530229239, 4474266830778691763689997443107122196, 157509566201473642198882091562701036336, 5701341163071319501511405331124460088514, 212036119997026430954729162366834150622279, 8096540881343491809728180651482911673248356, 317216217428868257560714338546265114691752873, 12743859259486651033006885706396735613399084418] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board (with the circular convention) made from, {0, 2, 3, 4} is the following rational function: 3 3 3 2 4 3 5 3 5 4 5 5 - (-1 + 3 u t - 7 u t - 6 u t + u t - 150 u t - 146 u t - 24 u t 6 5 17 20 20 20 19 20 18 20 + 246 u t + 4 t u + 25 t u + 68 t u + 34 t u 17 16 17 15 18 18 18 17 18 16 - 28 u t - 9 u t - 16 u t - 75 u t - 53 u t 18 15 19 19 19 17 19 16 19 15 - 8 u t + 12 u t - 64 u t - 34 u t - 4 u t 11 7 11 6 11 5 12 9 12 8 13 10 - 401 u t - 66 u t - 4 u t + 431 u t + 47 u t - 262 u t 13 9 13 8 13 7 13 13 13 12 - 195 u t - 50 u t - 4 u t + 36 u t + 121 u t 13 11 14 10 14 9 14 13 14 12 - 38 u t + 203 u t + 20 u t + 464 u t + 866 u t 14 11 15 10 15 9 15 15 15 13 + 654 u t + 50 u t + 4 u t - 51 u t + 256 u t 15 12 15 11 16 15 16 14 16 13 + 379 u t + 212 u t - 380 u t - 458 u t - 288 u t 16 12 16 11 17 17 9 6 11 8 - 84 u t - 8 u t - 9 u t + 864 u t - 1047 u t 12 10 12 11 12 12 5 8 3 8 4 + 1074 u t + 868 u t + 192 u t - 4 u t + 4 u t + 51 u t 9 3 9 4 9 5 10 5 10 6 14 14 + 4 u t + 66 u t + 375 u t - 20 u t - 251 u t + 51 t u 14 15 16 16 5 2 6 6 4 4 2 - 38 t u - 94 t u - 46 u t + 59 u t + 4 u t + 2 u t 8 8 4 2 7 6 7 7 6 3 6 4 - 80 t u - u t + 254 u t + 40 u t + 23 u t + 160 u t 7 3 7 4 7 5 8 5 8 6 8 7 + 13 u t + 127 u t + 338 u t + 181 u t + 109 u t - 173 u t 9 7 9 8 9 9 10 7 10 8 10 9 + 728 u t + 135 u t - 8 u t - 1070 u t - 1764 u t - 989 u t 10 10 11 9 11 10 11 11 3 19 18 - 140 u t - 1083 u t - 314 u t + 8 u t - u t - 6 u t ) / 3 3 2 2 3 2 5 3 5 4 / (1 - u - 4 u t + 4 u t + 3 t u + 2 u t + 8 u t + 14 u t / 5 5 6 5 9 6 11 8 12 10 12 11 12 12 + 4 u t - 16 u t - u t + u t - 2 u t - 4 u t - 4 u t 14 14 14 15 16 16 5 2 6 6 4 4 8 8 - t u - t u + t u + u t - 8 u t - 4 u t + 8 t u 7 6 7 7 6 4 7 4 7 5 8 6 8 7 - 15 u t - 4 u t - 4 u t - u t - 8 u t + u t + 12 u t 9 7 9 8 10 8 10 9 10 10 11 9 - 4 u t - 3 u t + 4 u t + 8 u t + 4 u t + 4 u t 11 10 + 4 u t ) and in Maple input form -(-1+3*u*t-7*u^3*t^3-6*u^3*t^2+u^4*t^3-150*u^5*t^3-146*u^5*t^4-24*u^5*t^5+246*u ^6*t^5+4*t^17*u^20+25*t^20*u^20+68*t^19*u^20+34*t^18*u^20-28*u^17*t^16-9*u^17*t ^15-16*u^18*t^18-75*u^18*t^17-53*u^18*t^16-8*u^18*t^15+12*u^19*t^19-64*u^19*t^ 17-34*u^19*t^16-4*u^19*t^15-401*u^11*t^7-66*u^11*t^6-4*u^11*t^5+431*u^12*t^9+47 *u^12*t^8-262*u^13*t^10-195*u^13*t^9-50*u^13*t^8-4*u^13*t^7+36*u^13*t^13+121*u^ 13*t^12-38*u^13*t^11+203*u^14*t^10+20*u^14*t^9+464*u^14*t^13+866*u^14*t^12+654* u^14*t^11+50*u^15*t^10+4*u^15*t^9-51*u^15*t^15+256*u^15*t^13+379*u^15*t^12+212* u^15*t^11-380*u^16*t^15-458*u^16*t^14-288*u^16*t^13-84*u^16*t^12-8*u^16*t^11-9* u^17*t^17+864*u^9*t^6-1047*u^11*t^8+1074*u^12*t^10+868*u^12*t^11+192*u^12*t^12-\ 4*u^5*t+4*u^8*t^3+51*u^8*t^4+4*u^9*t^3+66*u^9*t^4+375*u^9*t^5-20*u^10*t^5-251*u ^10*t^6+51*t^14*u^14-38*t^14*u^15-94*t^16*u^16-46*u^5*t^2+59*u^6*t^6+4*u^4*t^4+ 2*u^2*t-80*t^8*u^8-u^4*t^2+254*u^7*t^6+40*u^7*t^7+23*u^6*t^3+160*u^6*t^4+13*u^7 *t^3+127*u^7*t^4+338*u^7*t^5+181*u^8*t^5+109*u^8*t^6-173*u^8*t^7+728*u^9*t^7+ 135*u^9*t^8-8*u^9*t^9-1070*u^10*t^7-1764*u^10*t^8-989*u^10*t^9-140*u^10*t^10-\ 1083*u^11*t^9-314*u^11*t^10+8*u^11*t^11-u^3*t-6*u^19*t^18)/(1-u-4*u*t+4*u^3*t^3 +3*t^2*u^2+2*u^3*t^2+8*u^5*t^3+14*u^5*t^4+4*u^5*t^5-16*u^6*t^5-u^9*t^6+u^11*t^8 -2*u^12*t^10-4*u^12*t^11-4*u^12*t^12-t^14*u^14-t^14*u^15+t^16*u^16+u^5*t^2-8*u^ 6*t^6-4*u^4*t^4+8*t^8*u^8-15*u^7*t^6-4*u^7*t^7-4*u^6*t^4-u^7*t^4-8*u^7*t^5+u^8* t^6+12*u^8*t^7-4*u^9*t^7-3*u^9*t^8+4*u^10*t^8+8*u^10*t^9+4*u^10*t^10+4*u^11*t^9 +4*u^11*t^10) Just for fun the number of such permutations whose length is, 400, equals: 1155270332285520666800398196417460715079188127765890632639372060334821188014187\ 4174514647262741720403585706987736753591014622128837015674842303804643105306647\ 2765193256242818210051783634768722927760217927168405201638837117717573233861921\ 6800466537788953334354308465687664669025423536573661289549489274677885134659227\ 8958386302512321749821912318288161486349424067241772576650312479042944435366696\ 4716000895620378639809886567551031006323066756728282061700861249832389122511843\ 5084399611436582543097301163901967882039300097891786190274715655448046113786751\ 6940094952192697242391818412052107017886913836919343003129697811134138454621990\ 0545423087611587227266771694979347045630762590302563671439853353890939844076566\ 1462229620739177127272438912969741284977927483628053047233438065634521543155478\ 845690691391999921358022271859973039689860078154440094645447790769311235219714 Theorem No., 15, : Let , A(n), be the number of ways of reseating n diners around a round table in such a way that the number of seats, looking clockwise, that each diner moved, is never in the set, {0, 1, 2, 3, 4} Here are the first, 40, terms, starting at n=1, for the sake of OEIS [0, 0, 0, 0, 0, 1, 2, 49, 484, 6208, 79118, 1081313, 15610304, 238518181, 3850864416, 65598500129, 1177003136892, 22203823852849, 439598257630414, 9117748844458320, 197776095898147080, 4479171132922158213, 105749311074795459594, 2598770324359627927649, 66383290103770305444480, 1760251216489120748275545, 48391914159027137532742558, 1377657963370292694495775225, 40569314026776219126763298976, 1234485126421877951507713624960, 38777118629633847745046858932138, 1256202680185266044898703117243265, 41932743746637256328443711198317404, 1441091035502811154275687784996185061, 50948124259497137135375465336457288096, 1851549267690185733620735390157268309049, 69119507792662607427974600971719931532040, 2648668037144734021343020986316877954735481, 104119818645873461852781305796510051274240186, 4196130304288772990466853230376478916389613840] The generating function w.r.t., u, of the sequence of rook polynomials in the variable, t, for the board (with the circular convention) made from, {0, 1, 2, 3, 4} is the following rational function: 20 20 19 20 18 20 17 20 16 15 - (-1 + 41 t u + 100 t u + 42 t u + 4 t u - 800 u t 16 14 16 13 16 12 16 11 17 17 - 721 u t - 480 u t - 142 u t - 12 u t - 113 u t 17 16 17 15 17 14 17 13 18 18 - 189 u t + 98 u t + 76 u t + 8 u t - 48 u t 18 17 18 16 18 15 19 19 19 18 - 184 u t - 111 u t - 14 u t + 64 u t + 92 u t 19 17 19 16 19 15 8 6 9 3 9 4 - 52 u t - 38 u t - 4 u t + 792 u t + 4 u t + 78 u t 9 5 10 5 10 6 10 7 11 7 + 490 u t - 34 u t - 489 u t - 2334 u t - 432 u t 11 6 11 5 12 9 12 8 12 7 13 10 - 70 u t - 4 u t + 190 u t - 31 u t - 6 u t - 415 u t 13 9 13 8 13 7 13 11 14 10 - 290 u t - 62 u t - 4 u t + 245 u t + 317 u t 14 9 14 13 14 12 14 11 15 10 + 26 u t + 1572 u t + 2160 u t + 1254 u t + 54 u t 15 9 15 13 15 12 15 11 3 2 + 4 u t + 72 u t + 332 u t + 220 u t + 4 u t - 4 u t 3 3 4 2 4 3 4 4 5 2 5 3 - 4 u t - 18 u t - 32 u t - 6 u t - 58 u t - 222 u t 5 4 5 5 6 4 6 5 6 6 7 4 - 233 u t - 41 u t + 366 u t + 654 u t + 197 u t + 208 u t 7 5 7 6 8 8 9 6 9 7 9 8 + 546 u t + 323 u t - 144 u t + 1111 u t + 577 u t - 424 u t 9 9 10 8 10 9 10 10 11 8 - 204 u t - 4353 u t - 2868 u t - 494 u t - 1060 u t 11 9 11 10 11 11 12 10 12 11 - 804 u t + 186 u t + 242 u t + 1031 u t + 1356 u t 12 12 13 13 13 12 15 15 15 14 + 463 u t + 336 u t + 944 u t - 277 u t - 471 u t 2 4 5 6 3 7 3 7 7 8 3 + 2 u t - 2 u t - 4 u t + 46 u t + 20 u t + u t + 10 u t 8 4 8 5 2 2 14 14 16 16 / + 145 u t + 624 u t - 3 t u + 282 t u - 295 t u ) / (( / 10 10 9 8 8 8 8 7 6 6 7 5 6 5 5 5 u t - u t - 2 u t + 2 u t - 4 u t - u t - 5 u t + 2 u t 5 4 4 4 3 3 4 3 5 3 3 2 2 2 + 6 u t + 4 u t - 2 u t + 5 u t + 2 u t - 4 u t + 4 t u 3 6 6 2 2 2 4 3 5 5 - u t - 4 u t + 1 - u) (u t + 1 - u t - 2 t u - u t - u t + u t ) ) and in Maple input form -(-1+41*t^20*u^20+100*t^19*u^20+42*t^18*u^20+4*t^17*u^20-800*u^16*t^15-721*u^16 *t^14-480*u^16*t^13-142*u^16*t^12-12*u^16*t^11-113*u^17*t^17-189*u^17*t^16+98*u ^17*t^15+76*u^17*t^14+8*u^17*t^13-48*u^18*t^18-184*u^18*t^17-111*u^18*t^16-14*u ^18*t^15+64*u^19*t^19+92*u^19*t^18-52*u^19*t^17-38*u^19*t^16-4*u^19*t^15+792*u^ 8*t^6+4*u^9*t^3+78*u^9*t^4+490*u^9*t^5-34*u^10*t^5-489*u^10*t^6-2334*u^10*t^7-\ 432*u^11*t^7-70*u^11*t^6-4*u^11*t^5+190*u^12*t^9-31*u^12*t^8-6*u^12*t^7-415*u^ 13*t^10-290*u^13*t^9-62*u^13*t^8-4*u^13*t^7+245*u^13*t^11+317*u^14*t^10+26*u^14 *t^9+1572*u^14*t^13+2160*u^14*t^12+1254*u^14*t^11+54*u^15*t^10+4*u^15*t^9+72*u^ 15*t^13+332*u^15*t^12+220*u^15*t^11+4*u*t-4*u^3*t^2-4*u^3*t^3-18*u^4*t^2-32*u^4 *t^3-6*u^4*t^4-58*u^5*t^2-222*u^5*t^3-233*u^5*t^4-41*u^5*t^5+366*u^6*t^4+654*u^ 6*t^5+197*u^6*t^6+208*u^7*t^4+546*u^7*t^5+323*u^7*t^6-144*u^8*t^8+1111*u^9*t^6+ 577*u^9*t^7-424*u^9*t^8-204*u^9*t^9-4353*u^10*t^8-2868*u^10*t^9-494*u^10*t^10-\ 1060*u^11*t^8-804*u^11*t^9+186*u^11*t^10+242*u^11*t^11+1031*u^12*t^10+1356*u^12 *t^11+463*u^12*t^12+336*u^13*t^13+944*u^13*t^12-277*u^15*t^15-471*u^15*t^14+2*u ^2*t-2*u^4*t-4*u^5*t+46*u^6*t^3+20*u^7*t^3+u^7*t^7+10*u^8*t^3+145*u^8*t^4+624*u ^8*t^5-3*t^2*u^2+282*t^14*u^14-295*t^16*u^16)/(u^10*t^10-u^9*t^8-2*u^8*t^8+2*u^ 8*t^7-4*u^6*t^6-u^7*t^5-5*u^6*t^5+2*u^5*t^5+6*u^5*t^4+4*u^4*t^4-2*u^3*t^3+5*u^4 *t^3+2*u^5*t^3-4*u^3*t^2+4*t^2*u^2-u^3*t-4*u*t+1-u)/(u^6*t^6+1-u*t-2*t^2*u^2-u^ 2*t-u^4*t^3+u^5*t^5) Just for fun the number of such permutations whose length is, 400, equals: 4207289495670124269143962258808866000647757862387247498802789112662610757552492\ 6245361157064184539702958363978550324962226542658036032612445380978405760806159\ 5598902667240389018957551082757689950825741211304214294024255262737018582675432\ 4538276502578692592028399397062522619253257733739027822159733732480548690644346\ 3907711981677214532471608433816531969912118778999907362547438560226387707895823\ 5972792192022699849939174474005807664631538061236341697796610811707488599986287\ 9414939741877959295965848907057112363996305751787343670736455777052079508647478\ 9724445898214654236796359820090869589898871831136366471830665913941377256982094\ 1601713771230481649471498151212684770613123587179292549400754313234191589745740\ 4744091032510937048904681829956462392361197336770516623612585421594938062170593\ 72156026130328797595594605425461707594658864805838127385666270458889654436880 This ends this book, that took, 12348.191, seconds.