--------------------------------------- The generating functions for sequences enumerating Order-Ideals of P_{n+1,n+2} confined to the k outermost diagonals for k from 1 to, 20 By Shalosh B. Ekhad The generating function enumerating (n+1,n+2)-core partitions supported in t\ he , 1, outermost diagonals is 1 - ------- 2 t - 1 and in Maple format -1/(2*t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368, 68719476736, 137438953472, 274877906944, 549755813888, 1099511627776] The generating function enumerating (n+1,n+2)-core partitions supported in t\ he , 2, outermost diagonals is -1 + t - ------------ 2 t - 3 t + 1 and in Maple format -(-1+t)/(t^2-3*t+1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229 , 1346269, 3524578, 9227465, 24157817, 63245986, 165580141, 433494437, 1134903170, 2971215073, 7778742049, 20365011074, 53316291173, 139583862445, 365435296162, 956722026041, 2504730781961, 6557470319842, 17167680177565, 44945570212853, 117669030460994, 308061521170129, 806515533049393, 2111485077978050, 5527939700884757, 14472334024676221, 37889062373143906] The generating function enumerating (n+1,n+2)-core partitions supported in t\ he , 3, outermost diagonals is 2 t - 1 - -------------- 2 3 t - 4 t + 1 and in Maple format -(2*t-1)/(3*t^2-4*t+1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162, 2391485, 7174454, 21523361, 64570082, 193710245, 581130734, 1743392201, 5230176602, 15690529805, 47071589414, 141214768241, 423644304722, 1270932914165 , 3812798742494, 11438396227481, 34315188682442, 102945566047325, 308836698141974, 926510094425921, 2779530283277762, 8338590849833285, 25015772549499854, 75047317648499561, 225141952945498682, 675425858836496045, 2026277576509488134, 6078832729528464401] The generating function enumerating (n+1,n+2)-core partitions supported in t\ he , 4, outermost diagonals is 2 t - 3 t + 1 - ------------------- 3 2 t - 6 t + 5 t - 1 and in Maple format -(t^2-3*t+1)/(t^3-6*t^2+5*t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 131, 417, 1341, 4334, 14041, 45542, 147798, 479779, 1557649, 5057369, 16420730, 53317085, 173118414, 562110290, 1825158051, 5926246929, 19242396629, 62479659622, 202870165265, 658715265222, 2138834994142, 6944753544643, 22549473023585, 73217678844209, 237736309624178, 771924948079221 , 2506424561495246, 8138309428625082, 26424924722233155, 85801191600910529, 278594719099778797, 904591370615663966, 2937189730080557577, 9536995145808582886, 30966428719175232934, 100547362451105224931] The generating function enumerating (n+1,n+2)-core partitions supported in t\ he , 5, outermost diagonals is 2 3 t - 4 t + 1 - ---------------------- 3 2 4 t - 10 t + 6 t - 1 and in Maple format -(3*t^2-4*t+1)/(4*t^3-10*t^2+6*t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 428, 1416, 4744, 16016, 54320, 184736, 629280, 2145600, 7319744, 24979584, 85262464, 291057920, 993641216, 3392317952, 11581727232, 39541748736, 135002491904, 460924372992, 1573688313856, 5372896120832, 18344191078400, 62630938517504, 213835304804352, 730079207964672, 2492645953814528, 8510424862457856, 29056406468460544, 99204774001442816, 338706278773882880, 1156415558502711296, 3948249659283210240, 13480167485767680000, 46024170555784822784, 157136347114164977664, 536497047070212358144] The generating function enumerating (n+1,n+2)-core partitions supported in t\ he , 6, outermost diagonals is 3 2 t - 6 t + 5 t - 1 - ---------------------------- 4 3 2 t - 10 t + 15 t - 7 t + 1 and in Maple format -(t^3-6*t^2+5*t-1)/(t^4-10*t^3+15*t^2-7*t+1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1429, 4846, 16645, 57686, 201158, 704420, 2473785, 8704089, 30664890, 108126325, 381478030, 1346396146, 4753200932, 16783118309, 59266297613, 209302921830, 739203970773, 2610763825782, 9221050139566, 32568630376132, 115033094826481, 406301945713265, 1435082451217394, 5068810290711461, 17903421629025486, 63236319608967162, 223355933283285060, 788912144848031389, 2786505789147589957, 9842181364536542302, 34763448229738975589, 122787562886752562774, 433696524617401138646, 1531854529953372742500] The generating function enumerating (n+1,n+2)-core partitions supported in t\ he , 7, outermost diagonals is 3 2 4 t - 10 t + 6 t - 1 - ------------------------------ 4 3 2 5 t - 20 t + 21 t - 8 t + 1 and in Maple format -(4*t^3-10*t^2+6*t-1)/(5*t^4-20*t^3+21*t^2-8*t+1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4861, 16778, 58598, 206516, 732825, 2613834, 9358677, 33602822, 120902914, 435668420, 1571649221, 5674201118, 20497829133, 74079051906, 267803779710, 968355724724, 3502058316337, 12666676646162, 45818284122149, 165745451110910, 599602883663706, 2169200895192708, 7847764206211293, 28392225268363046, 102720757302002845, 371638607428661114, 1344578640423433622, 4864672387083827252, 17600396009841720009, 63678427721298754650, 230389660103273293941, 833554856940327767798, 3015822567730649462578] The generating function enumerating (n+1,n+2)-core partitions supported in t\ he , 8, outermost diagonals is 4 3 2 t - 10 t + 15 t - 7 t + 1 - ------------------------------------ 5 4 3 2 t - 15 t + 35 t - 28 t + 9 t - 1 and in Maple format -(t^4-10*t^3+15*t^2-7*t+1)/(t^5-15*t^4+35*t^3-28*t^2+9*t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16795, 58766, 207783, 740924, 2660139, 9603089, 34818270, 126676726, 462125928, 1689438278, 6186432967, 22682699779, 83249302471, 305773834030, 1123771473120, 4131947428007, 15197952958467, 55915691993228, 205765906819451, 757326699164035, 2787706774345669, 10262482704258873, 37782416217093813, 139107832756053214, 512191460458891717, 1885938860769281490, 6944389239984279502, 25571081099256139496, 94160927226542959017, 346734806208084177202, 1276815232264524713662, 4701773142234511213959] The generating function enumerating (n+1,n+2)-core partitions supported in t\ he , 9, outermost diagonals is 4 3 2 5 t - 20 t + 21 t - 8 t + 1 - --------------------------------------- 5 4 3 2 6 t - 35 t + 56 t - 36 t + 10 t - 1 and in Maple format -(5*t^4-20*t^3+21*t^2-8*t+1)/(6*t^5-35*t^4+56*t^3-36*t^2+10*t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58785, 207990, 742626, 2671892, 9675125, 35223254, 128810022, 472809612, 1740839529, 6425847350, 23768577482, 88066807556, 326754211965, 1213717766550, 4512422130126, 16788890775932, 62501909428145, 232795069077494, 867407366858802, 3233021465397492, 12053239830068229, 44945622129322070, 167626302047430902, 625250747214775916, 2332450172422279545, 8701770403601129910, 32466292833952537626, 121138385125299782372, 452012200280878615565, 1686685273907830227734, 6294053469216727892382] The generating function enumerating (n+1,n+2)-core partitions supported in t\ he , 10, outermost diagonals is 5 4 3 2 t - 15 t + 35 t - 28 t + 9 t - 1 - --------------------------------------------- 6 5 4 3 2 t - 21 t + 70 t - 84 t + 45 t - 11 t + 1 and in Maple format -(t^5-15*t^4+35*t^3-28*t^2+9*t-1)/(t^6-21*t^5+70*t^4-84*t^3+45*t^2-11*t+1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208011, 742876, 2674117, 9691625, 35331134, 129453302, 476385987, 1759662354, 6520664475, 24229457252, 90241935251, 336769814966, 1258885297696, 4712531446251, 17662034439832, 66261792898720, 248800766918879, 934861268810237, 3514791913340867, 13221083060354529, 49752406075732470, 187288515333417342, 705234224273645456, 2656195342054378046, 10006303485344001056, 37701606156514031251, 142071330330512901897, 535430805612013277290, 2018097537991032309499, 7607042160018029142347] The generating function enumerating (n+1,n+2)-core partitions supported in t\ he , 11, outermost diagonals is 5 4 3 2 6 t - 35 t + 56 t - 36 t + 10 t - 1 - ------------------------------------------------- 6 5 4 3 2 7 t - 56 t + 126 t - 120 t + 55 t - 12 t + 1 and in Maple format -(6*t^5-35*t^4+56*t^3-36*t^2+10*t-1)/(7*t^6-56*t^5+126*t^4-120*t^3+55*t^2-12*t+ 1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742899, 2674414, 9694469, 35353670, 129609821, 477372378, 1765432578, 6552495612, 24396947705, 91089777818, 340925763074, 1278713272108, 4804981949782, 18084694597452, 68161523360923, 257214253446866, 971644345429865, 3673787485895594, 13901477326061013, 52638194151782694, 199431255200842938, 755966023536488180, 2866800990313876421, 10875584968094347838, 41270978350158746086, 156658649760968207452, 594791337796298203722, 2258713106034310399852, 8578893544092592824899] The generating function enumerating (n+1,n+2)-core partitions supported in t\ he , 12, outermost diagonals is 6 5 4 3 2 t - 21 t + 70 t - 84 t + 45 t - 11 t + 1 - -------------------------------------------------------- 7 6 5 4 3 2 t - 28 t + 126 t - 210 t + 165 t - 66 t + 13 t - 1 and in Maple format -(t^6-21*t^5+70*t^4-84*t^3+45*t^2-11*t+1)/(t^7-28*t^6+126*t^5-210*t^4+165*t^3-\ 66*t^2+13*t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674439, 9694817, 35357237, 129639894, 477593445, 1766900442, 6561506666, 24448929596, 91374913097, 342426238270, 1286340295714, 4842631785292, 18265961098742, 69015665738684, 261164661944060, 989619890402247, 3754420405973361, 14258635675500681, 54202589499479882, 206215473636693365, 785125362836536214, 2991131011295570842, 11401892498430894772, 43484430036843244549, 165912712058533136150, 633273589243380018902, 2417956408446856313756, 9234915513515790234666] The generating function enumerating (n+1,n+2)-core partitions supported in t\ he , 13, outermost diagonals is 6 5 4 3 2 7 t - 56 t + 126 t - 120 t + 55 t - 12 t + 1 - ---------------------------------------------------------- 7 6 5 4 3 2 8 t - 84 t + 252 t - 330 t + 220 t - 78 t + 14 t - 1 and in Maple format -(7*t^6-56*t^5+126*t^4-120*t^3+55*t^2-12*t+1)/(8*t^7-84*t^6+252*t^5-330*t^4+220 *t^3-78*t^2+14*t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694844, 35357640, 129644296, 477632784, 1767205544, 6563635312, 24462610640, 91457275168, 342896363376, 1288908109600, 4856148112032, 18334910316352, 69358059019424, 262825903417280, 997518644250688, 3791317346771584, 14428320101827008, 54972229576061056, 209663639795423360, 800404935106760960, 3058173073078870656, 11693449237355423488, 44742227717209810176, 171299591524206912000, 656192313239394113280, 2514879004907517809152, 9642546533834587445760] The generating function enumerating (n+1,n+2)-core partitions supported in t\ he , 14, outermost diagonals is 7 6 5 4 3 2 t - 28 t + 126 t - 210 t + 165 t - 66 t + 13 t - 1 - ----------------------------------------------------------------- 8 7 6 5 4 3 2 t - 36 t + 210 t - 462 t + 495 t - 286 t + 91 t - 15 t + 1 and in Maple format -(t^7-28*t^6+126*t^5-210*t^4+165*t^3-66*t^2+13*t-1)/(t^8-36*t^7+210*t^6-462*t^5 +495*t^4-286*t^3+91*t^2-15*t+1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357669, 129644758, 477638141, 1767256122, 6564048010, 24465628732, 91477540853, 343023451854, 1289661784962, 4860414840652, 18358138660582, 69480371553404, 263451765822140, 1000642532381128, 3806574971452729, 14501432976764089, 55316720239735242, 211262661377872805, 807728513712307350, 3091315673573407706, 11841825961843658676, 45400060554224782749, 174190497458426156950, 668795233051773102230, 2569421686856186575004, 9877025660653679630922] The generating function enumerating (n+1,n+2)-core partitions supported in t\ he , 15, outermost diagonals is 7 6 5 4 3 2 8 t - 84 t + 252 t - 330 t + 220 t - 78 t + 14 t - 1 - --------------------------------------------------------------------- 8 7 6 5 4 3 2 9 t - 120 t + 462 t - 792 t + 715 t - 364 t + 105 t - 16 t + 1 and in Maple format -(8*t^7-84*t^6+252*t^5-330*t^4+220*t^3-78*t^2+14*t-1)/(9*t^8-120*t^7+462*t^6-\ 792*t^5+715*t^4-364*t^3+105*t^2-16*t+1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644789, 477638666, 1767262562, 6564112060, 24466177189, 91481735890, 343052825154, 1289853348972, 4861593400902, 18365045288492, 69519215609564, 263662687669640, 1001753626952953, 3812275656229354, 14530013464239493, 55457120418290886, 211940030660378106, 810944384733610356, 3106365135385964381, 11911348611991034318, 45717503144106794726, 175624726027181487964, 675213352140403788394, 2597892859347924681004, 10002322574722773980100] The generating function enumerating (n+1,n+2)-core partitions supported in t\ he , 16, outermost diagonals is 8 7 6 5 4 3 2 / 9 - (t - 36 t + 210 t - 462 t + 495 t - 286 t + 91 t - 15 t + 1) / (t / 8 7 6 5 4 3 2 - 45 t + 330 t - 924 t + 1287 t - 1001 t + 455 t - 120 t + 17 t - 1 ) and in Maple format -(t^8-36*t^7+210*t^6-462*t^5+495*t^4-286*t^3+91*t^2-15*t+1)/(t^9-45*t^8+330*t^7 -924*t^6+1287*t^5-1001*t^4+455*t^3-120*t^2+17*t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638699, 1767263154, 6564119719, 24466257220, 91482453431, 343058554086, 1289895105840, 4861876179692, 18366847744602, 69530136810764, 263726076141605, 1002108234455488, 3814197149470363, 14540139792907249, 55509198680010774, 212202141534223014, 812238525898058048, 3112646261414447198, 11941369177101086832, 45859013696222419707, 176283482045650829386, 678245366658557440502, 2611704746636463934580, 10064650426403627944359] The generating function enumerating (n+1,n+2)-core partitions supported in t\ he , 17, outermost diagonals is 8 7 6 5 4 3 2 / - (9 t - 120 t + 462 t - 792 t + 715 t - 364 t + 105 t - 16 t + 1) / ( / 9 8 7 6 5 4 3 2 10 t - 165 t + 792 t - 1716 t + 2002 t - 1365 t + 560 t - 136 t + 18 t - 1) and in Maple format -(9*t^8-120*t^7+462*t^6-792*t^5+715*t^4-364*t^3+105*t^2-16*t+1)/(10*t^9-165*t^8 +792*t^7-1716*t^6+2002*t^5-1365*t^4+560*t^3-136*t^2+18*t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263189, 6564120382, 24466266242, 91482552244, 343059479790, 1289902806884, 4861934517972, 18367257405672, 69532838824169, 263742984852710, 1002209418119602, 3814779806879476, 14543384909816221, 55526752851860262, 212294691668475590, 812715498576342540, 3115055112654979610, 11953315948849283260, 45917306020579460820, 176563755958125237000, 679575105822355150869, 2617937651250826159782, 10093545223689742556302] The generating function enumerating (n+1,n+2)-core partitions supported in t\ he , 18, outermost diagonals is 9 8 7 6 5 4 3 2 - (t - 45 t + 330 t - 924 t + 1287 t - 1001 t + 455 t - 120 t + 17 t / 10 9 8 7 6 5 4 - 1) / (t - 55 t + 495 t - 1716 t + 3003 t - 3003 t + 1820 t / 3 2 - 680 t + 153 t - 19 t + 1) and in Maple format -(t^9-45*t^8+330*t^7-924*t^6+1287*t^5-1001*t^4+455*t^3-120*t^2+17*t-1)/(t^10-55 *t^9+495*t^8-1716*t^7+3003*t^6-3003*t^5+1820*t^4-680*t^3+153*t^2-19*t+1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120419, 24466266980, 91482562781, 343059600494, 1289903986208, 4861944723692, 18367337640827, 69533422307264, 263746962622649, 1002235100697952, 3814938115729213, 14544322519180009, 55532116380693974, 212324453204410022, 812876263295858276, 3115903055893628398, 11957694164599879072, 45939484732224163708, 176674191278504852620, 680116517590969507510, 2620554752019696530500, 10106034438694020866819] The generating function enumerating (n+1,n+2)-core partitions supported in t\ he , 19, outermost diagonals is 9 8 7 6 5 4 3 2 - (10 t - 165 t + 792 t - 1716 t + 2002 t - 1365 t + 560 t - 136 t / 10 9 8 7 6 5 + 18 t - 1) / (11 t - 220 t + 1287 t - 3432 t + 5005 t - 4368 t / 4 3 2 + 2380 t - 816 t + 171 t - 20 t + 1) and in Maple format -(10*t^9-165*t^8+792*t^7-1716*t^6+2002*t^5-1365*t^4+560*t^3-136*t^2+18*t-1)/(11 *t^10-220*t^9+1287*t^8-3432*t^7+5005*t^6-4368*t^5+2380*t^4-816*t^3+171*t^2-20*t +1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267019, 91482563598, 343059612706, 1289904132236, 4861946209127, 18367350992542, 69533531097608, 263747780950016, 1002240860726875, 3814976449984414, 14544565712377173, 55533596687111494, 212333144717804318, 812925704417191100, 3116176533926394222, 11959169672995193908, 45947270304957332440, 176714458493885570480, 680321055658813457350, 2621576860871871947220, 10111066745690230028775] The generating function enumerating (n+1,n+2)-core partitions supported in t\ he , 20, outermost diagonals is 10 9 8 7 6 5 4 3 - (t - 55 t + 495 t - 1716 t + 3003 t - 3003 t + 1820 t - 680 t 2 / 11 10 9 8 7 + 153 t - 19 t + 1) / (t - 66 t + 715 t - 3003 t + 6435 t / 6 5 4 3 2 - 8008 t + 6188 t - 3060 t + 969 t - 190 t + 21 t - 1) and in Maple format -(t^10-55*t^9+495*t^8-1716*t^7+3003*t^6-3003*t^5+1820*t^4-680*t^3+153*t^2-19*t+ 1)/(t^11-66*t^10+715*t^9-3003*t^8+6435*t^7-8008*t^6+6188*t^5-3060*t^4+969*t^3-\ 190*t^2+21*t-1) For the sake of the OEIS, here are the first, 41, terms, starting with n=0 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563639, 343059613606, 1289904146291, 4861946384252, 18367352844301, 69533548361192, 263747926554266, 1002241992345880, 3814984665821439, 14544622023036577, 55533964058837038, 212335441276414118, 812939535487775540, 3116257143017528622, 11959626013187137156, 45949787663227009948, 176728026810588468980, 680392676606686617750, 2621947849394859905095, 10112955815268790044660] This ends this article, that took, 0.104, seconds to generate.