Rational generating functions for the sequences enumerating (n+1,n+2) Core P\ artitions into Odd parts whose Corresponding Order-Ideals have width is <=r for r from 1 to, 5, and their first, 30, terms. By Shalosh B. Ekhad For the width, 1, the generating function is t + 1 - ---------- 2 t + t - 1 and in Maple format -(t+1)/(t^2+t-1) The first, 31, terms, staring with n=0, are [1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040 , 1346269, 2178309] For the width, 2, the generating function is 4 3 2 t - t - t + t + 1 - ----------------------------- 5 4 3 2 t - t - 2 t + 3 t + t - 1 and in Maple format -(t^4-t^3-t^2+t+1)/(t^5-t^4-2*t^3+3*t^2+t-1) The first, 31, terms, staring with n=0, are [1, 2, 4, 7, 15, 27, 56, 104, 210, 398, 791, 1517, 2988, 5769, 11306, 21911, 42820, 83160, 162261, 315496, 615050, 1196676, 2331733, 4538426, 8840719, 17210905, 33521153, 65265737, 127105093, 247489812, 481963369] For the width, 3, the generating function is 9 8 7 6 5 4 3 2 t + t - 4 t - 6 t + 8 t + 9 t - 5 t - 5 t + t + 1 - ---------------------------------------------------------------- 9 8 7 6 5 4 3 2 (t + 2 t - 3 t - 9 t + 3 t + 14 t - t - 7 t + 1) (t - 1) and in Maple format -(t^9+t^8-4*t^7-6*t^6+8*t^5+9*t^4-5*t^3-5*t^2+t+1)/(t^9+2*t^8-3*t^7-9*t^6+3*t^5 +14*t^4-t^3-7*t^2+1)/(t-1) The first, 31, terms, staring with n=0, are [1, 2, 4, 7, 17, 31, 76, 144, 344, 670, 1560, 3103, 7079, 14315, 32152, 65861, 146183, 302456, 665300, 1387172, 3030464, 6356068, 13813464, 29103412, 62999146 , 133190358, 287443371, 609299853, 1311936956, 2786508393, 5989399832] For the width, 4, the generating function is 20 19 18 17 16 15 14 13 12 - (t - t - 9 t + 9 t + 41 t - 44 t - 118 t + 132 t + 234 t 11 10 9 8 7 6 5 4 - 246 t - 330 t + 279 t + 319 t - 184 t - 194 t + 68 t + 69 t 3 2 / 21 20 19 18 17 - 13 t - 13 t + t + 1) / (t - t - 10 t + 11 t + 49 t / 16 15 14 13 12 11 10 - 61 t - 150 t + 208 t + 308 t - 452 t - 432 t + 627 t 9 8 7 6 5 4 3 2 + 403 t - 547 t - 234 t + 292 t + 79 t - 91 t - 14 t + 15 t + t - 1) and in Maple format -(t^20-t^19-9*t^18+9*t^17+41*t^16-44*t^15-118*t^14+132*t^13+234*t^12-246*t^11-\ 330*t^10+279*t^9+319*t^8-184*t^7-194*t^6+68*t^5+69*t^4-13*t^3-13*t^2+t+1)/(t^21 -t^20-10*t^19+11*t^18+49*t^17-61*t^16-150*t^15+208*t^14+308*t^13-452*t^12-432*t ^11+627*t^10+403*t^9-547*t^8-234*t^7+292*t^6+79*t^5-91*t^4-14*t^3+15*t^2+t-1) The first, 31, terms, staring with n=0, are [1, 2, 4, 7, 17, 31, 80, 152, 396, 774, 1996, 3983, 10113, 20519, 51292, 105549 , 260149, 541936, 1319356, 2778052, 6691524, 14222324, 33944196, 72737388, 172233372, 371700702, 874170060, 1898220191, 4438175543, 9688793827, 22539201400] For the width, 5, the generating function is 41 40 39 38 37 36 35 34 - (t + t - 20 t - 22 t + 200 t + 241 t - 1321 t - 1717 t 33 32 31 30 29 28 + 6413 t + 8805 t - 24116 t - 34182 t + 72196 t + 103230 t 27 26 25 24 23 - 174298 t - 246110 t + 340608 t + 466664 t - 537608 t 22 21 20 19 18 - 705960 t + 681448 t + 852171 t - 688151 t - 818891 t 17 16 15 14 13 + 548287 t + 623263 t - 340760 t - 372562 t + 162944 t 12 11 10 9 8 7 + 172730 t - 58906 t - 61022 t + 15715 t + 16019 t - 2984 t 6 5 4 3 2 / 40 38 - 3010 t + 380 t + 381 t - 29 t - 29 t + t + 1) / ((t - 20 t / 37 36 35 34 33 32 31 - 2 t + 202 t + 39 t - 1360 t - 359 t + 6774 t + 2068 t 30 29 28 27 26 25 - 26219 t - 8287 t + 80772 t + 24237 t - 200025 t - 52882 t 24 23 22 21 20 + 398797 t + 86797 t - 638028 t - 107191 t + 814275 t 19 18 17 16 15 14 + 99057 t - 822733 t - 67824 t + 652252 t + 33927 t - 401462 t 13 12 11 10 9 8 - 12149 t + 189367 t + 3016 t - 67301 t - 491 t + 17597 t 7 6 5 4 2 2 + 47 t - 3265 t - 2 t + 405 t - 30 t + 1) (t + t - 1)) and in Maple format -(t^41+t^40-20*t^39-22*t^38+200*t^37+241*t^36-1321*t^35-1717*t^34+6413*t^33+ 8805*t^32-24116*t^31-34182*t^30+72196*t^29+103230*t^28-174298*t^27-246110*t^26+ 340608*t^25+466664*t^24-537608*t^23-705960*t^22+681448*t^21+852171*t^20-688151* t^19-818891*t^18+548287*t^17+623263*t^16-340760*t^15-372562*t^14+162944*t^13+ 172730*t^12-58906*t^11-61022*t^10+15715*t^9+16019*t^8-2984*t^7-3010*t^6+380*t^5 +381*t^4-29*t^3-29*t^2+t+1)/(t^40-20*t^38-2*t^37+202*t^36+39*t^35-1360*t^34-359 *t^33+6774*t^32+2068*t^31-26219*t^30-8287*t^29+80772*t^28+24237*t^27-200025*t^ 26-52882*t^25+398797*t^24+86797*t^23-638028*t^22-107191*t^21+814275*t^20+99057* t^19-822733*t^18-67824*t^17+652252*t^16+33927*t^15-401462*t^14-12149*t^13+ 189367*t^12+3016*t^11-67301*t^10-491*t^9+17597*t^8+47*t^7-3265*t^6-2*t^5+405*t^ 4-30*t^2+1)/(t^2+t-1) The first, 31, terms, staring with n=0, are [1, 2, 4, 7, 17, 31, 80, 152, 404, 790, 2124, 4239, 11393, 23095, 61664, 126613 , 334969, 695464, 1821872, 3820340, 9911724, 20972908, 53920336, 115043812, 293289848, 630554510, 1595122108, 3453609519, 8674934161, 18904149423, 47177630716] This ends this article, that took, 159.915, seconds to generate.