1153:
955:
2154:
Given the lack of an explicitly known generating function, the enumerations of the numbers of solid partitions for larger integers have been carried out numerically. There are two algorithms that are used to enumerate solid partitions and their higher-dimensional generalizations. The work of Atkin.
1148:{\displaystyle \left({\begin{smallmatrix}0\\0\\0\\0\end{smallmatrix}}{\begin{smallmatrix}0\\0\\1\\0\end{smallmatrix}}{\begin{smallmatrix}0\\1\\0\\0\end{smallmatrix}}{\begin{smallmatrix}1\\0\\0\\0\end{smallmatrix}}{\begin{smallmatrix}1\\1\\0\\0\end{smallmatrix}}\right)\ ,}
390:
2120:
2145:
fails to correctly reproduce the solid partitions of 6. It appears that there is no simple formula for the generating function of solid partitions; in particular, there cannot be any formula analogous to the product formulas of Euler and MacMahon.
628:
2406:
1799:
773:
680:
492:
in notation where plane partitions are two-dimensional partitions and partitions are one-dimensional partitions. Solid partitions and their higher-dimensional generalizations are discussed in the book by
2155:
et al. used an algorithm due to
Bratley and McKay. In 1970, Knuth proposed a different algorithm to enumerate topological sequences that he used to evaluate numbers of solid partitions of all integers
853:
208:
2282:
1935:
1627:
197:
899:
2545:
D P Bhatia, M A Prasad and D Arora, "Asymptotic results for the number of multidimensional partitions of an integer and directed compact lattice animals", J. Phys. A: Math. Gen.
1569:
427:
1891:
1316:
127:
1841:
1500:
1461:
1418:
1375:
1254:
943:
89:
2231:
2205:
2179:
1927:
466:
2320:
1205:
on a
Ferrers diagram – this corresponds to permuting the four coordinates of all nodes. This generalises the operation denoted by conjugation on usual partitions.
1203:
2313:
1651:
1336:
1176:
547:
527:
486:
50:
556:
2515:
Srivatsan
Balakrishnan, Suresh Govindarajan and Naveen S. Prabhakar, "On the asymptotics of higher-dimensional partitions", J.Phys. A: Math. Gen.
2499:
Ville
Mustonen and R. Rajesh, "Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer", J. Phys. A: Math. Gen.
2207:. In 2010, S. Balakrishnan proposed a parallel version of Knuth's algorithm that has been used to extend the enumeration to all integers
1659:
692:
2590:
2423:
P. A. MacMahon, Combinatory
Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
633:
2478:
P. Bratley and J. K. S. McKay, "Algorithm 313: Multi-dimensional partition generator", Comm. ACM, 10 (Issue 10, 1967), p. 666.
494:
778:
385:{\displaystyle n_{i+1,j,k}\leq n_{i,j,k},\quad n_{i,j+1,k}\leq n_{i,j,k}\quad {\text{and}}\quad n_{i,j,k+1}\leq n_{i,j,k}}
1377:
form a solid partition. One can verify that condition FD implies that the conditions for a solid partition are satisfied.
2595:
2239:
2115:{\displaystyle P_{3}(q):=\sum _{n=0}^{\infty }p_{3}(n)q^{n}=1+q+4q^{2}+10q^{3}+26q^{4}+59q^{5}+140q^{6}+\cdots .}
2142:
2138:
29:
1574:
135:
858:
488:. As the definition of solid partitions involves three-dimensional arrays of numbers, they are also called
506:
1505:
395:
1854:
1259:
1213:
Given a
Ferrers diagram, one constructs the solid partition (as in the main definition) as follows.
2532:
Destainville, N., & Govindarajan, S. (2015). Estimating the asymptotics of solid partitions.
2461:
94:
2287:
which is a 19 digit number illustrating the difficulty in carrying out such exact enumerations.
1807:
1466:
1427:
1384:
1341:
1220:
904:
55:
2210:
2184:
2158:
2126:
1896:
435:
21:
1181:
2130:
1629:
to the
Ferrers diagram. By construction, it is easy to see that condition FD is satisfied.
25:
2298:
2134:
1636:
1420:
that form a solid partition, one obtains the corresponding
Ferrers diagram as follows.
1321:
1161:
532:
512:
471:
35:
2584:
623:{\displaystyle \lambda =(\mathbf {y} _{1},\mathbf {y} _{2},\ldots ,\mathbf {y} _{n})}
2448:
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for
2575:
2570:
1102:
1068:
1034:
1000:
966:
2401:{\displaystyle \lim _{n\rightarrow \infty }{\frac {\log p_{3}(n)}{n^{3/4}}}=c.}
2487:
D. E. Knuth, "A note on solid partitions", Math. Comp., 24 (1970), 955–961.
1256:
be the number of nodes in the
Ferrers diagram with coordinates of the form
2504:
2566:
sequence A000293 (Solid (i.e., three-dimensional) partitions)
2451:
m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097–1100.
1794:{\displaystyle n_{1,1,1}=n_{2,1,1}=n_{1,2,1}=n_{1,1,2}=n_{2,2,1}=1}
2520:
768:{\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3},a_{4})\in \lambda }
2562:
1424:
Start with the
Ferrers diagram with no nodes. For every non-zero
2181:. Mustonen and Rajesh extended the enumeration for all integers
1158:
where each column is a node, represents a solid partition of
505:
Another representation for solid partitions is in the form of
675:{\displaystyle \mathbf {y} _{i}\in \mathbb {Z} _{\geq 0}^{4}}
1653:
nodes given above corresponds to the solid partition with
2565:
1893:. Define the generating function of solid partitions,
848:{\displaystyle \mathbf {y} =(y_{1},y_{2},y_{3},y_{4})}
509:
diagrams. The Ferrers diagram of a solid partition of
52:
is a three-dimensional array of non-negative integers
2323:
2301:
2242:
2213:
2187:
2161:
1938:
1899:
1857:
1810:
1662:
1639:
1577:
1508:
1469:
1430:
1387:
1344:
1324:
1262:
1223:
1184:
1178:. There is a natural action of the permutation group
1164:
958:
907:
861:
781:
695:
636:
559:
535:
515:
474:
438:
398:
211:
138:
97:
58:
38:
2400:
2307:
2276:
2225:
2199:
2173:
2114:
1921:
1885:
1835:
1793:
1645:
1621:
1563:
1494:
1455:
1412:
1369:
1330:
1310:
1248:
1197:
1170:
1147:
937:
893:
847:
767:
674:
622:
541:
521:
480:
460:
421:
384:
191:
121:
83:
44:
2325:
2277:{\displaystyle p_{3}(72)=3464274974065172792\ ,}
2295:It is conjectured that there exists a constant
8:
2444:
2442:
1338:denotes an arbitrary value. The collection
2571:The Solid Partitions Project of IIT Madras
2495:
2493:
2468:. Cambridge University Press. p. 402.
2377:
2373:
2353:
2340:
2328:
2322:
2300:
2247:
2241:
2212:
2186:
2160:
2097:
2081:
2065:
2049:
2033:
2005:
1986:
1976:
1965:
1943:
1937:
1904:
1898:
1862:
1856:
1815:
1809:
1767:
1742:
1717:
1692:
1667:
1661:
1638:
1601:
1588:
1576:
1552:
1507:
1474:
1468:
1435:
1429:
1392:
1386:
1349:
1343:
1323:
1261:
1228:
1222:
1189:
1183:
1163:
1100:
1066:
1032:
998:
964:
957:
906:
885:
872:
860:
836:
823:
810:
797:
782:
780:
750:
737:
724:
711:
696:
694:
666:
658:
654:
653:
643:
638:
635:
611:
606:
590:
585:
575:
570:
558:
534:
514:
473:
468:denote the number of solid partitions of
443:
437:
408:
397:
364:
333:
323:
304:
273:
247:
216:
210:
165:
143:
137:
96:
63:
57:
37:
2576:The Mathworld entry for Solid Partitions
1622:{\displaystyle 0\leq y_{4}<n_{i,j,k}}
192:{\displaystyle \sum _{i,j,k}n_{i,j,k}=n}
2416:
1633:For example, the Ferrers diagram with
1209:Equivalence of the two representations
2133:have simple product formulae, due to
894:{\displaystyle 0\leq y_{i}\leq a_{i}}
501:Ferrers diagrams for solid partitions
7:
2141:, respectively. However, a guess of
2466:Enumerative Combinatorics, volume 2
2436:, Cambridge University Press, 1998.
1564:{\displaystyle (i-1,j-1,k-1,y_{4})}
422:{\displaystyle i,j{\text{ and }}k.}
2335:
1977:
949:For instance, the Ferrers diagram
14:
2150:Exact enumeration using computers
1101:
1067:
1033:
999:
965:
1886:{\displaystyle p_{3}(0)\equiv 1}
783:
697:
639:
607:
586:
571:
1311:{\displaystyle (i-1,j-1,k-1,*)}
328:
322:
268:
20:are natural generalizations of
2534:Journal of Statistical Physics
2365:
2359:
2332:
2259:
2253:
1998:
1992:
1955:
1949:
1916:
1910:
1874:
1868:
1558:
1509:
1305:
1263:
842:
790:
756:
704:
617:
566:
455:
449:
1:
2125:The generating functions of
490:three-dimensional partitions
775:, then so do all the nodes
122:{\displaystyle i,j,k\geq 1}
2612:
682:satisfying the condition:
2591:Enumerative combinatorics
1836:{\displaystyle n_{i,j,k}}
1495:{\displaystyle n_{i,j,k}}
1456:{\displaystyle n_{i,j,k}}
1413:{\displaystyle n_{i,j,k}}
1370:{\displaystyle n_{i,j,k}}
1249:{\displaystyle n_{i,j,k}}
938:{\displaystyle i=1,2,3,4}
84:{\displaystyle n_{i,j,k}}
2434:The theory of partitions
2226:{\displaystyle n\leq 72}
2200:{\displaystyle n\leq 50}
2174:{\displaystyle n\leq 28}
1922:{\displaystyle P_{3}(q)}
461:{\displaystyle p_{3}(n)}
30:Percy Alexander MacMahon
32:. A solid partition of
2402:
2309:
2278:
2227:
2201:
2175:
2116:
1981:
1923:
1887:
1837:
1795:
1647:
1623:
1565:
1496:
1457:
1414:
1371:
1332:
1312:
1250:
1199:
1172:
1149:
939:
895:
849:
769:
676:
624:
543:
523:
482:
462:
423:
386:
193:
123:
85:
46:
2503:(2003), no. 24, 6651.
2403:
2310:
2279:
2228:
2202:
2176:
2117:
1961:
1924:
1888:
1838:
1796:
1648:
1624:
1566:
1497:
1458:
1415:
1372:
1333:
1313:
1251:
1200:
1198:{\displaystyle S_{4}}
1173:
1150:
940:
896:
850:
770:
677:
625:
544:
524:
483:
463:
424:
387:
194:
124:
86:
47:
2321:
2299:
2240:
2211:
2185:
2159:
1936:
1897:
1855:
1808:
1660:
1637:
1575:
1506:
1467:
1428:
1385:
1342:
1322:
1260:
1221:
1182:
1162:
956:
905:
859:
779:
693:
634:
557:
533:
513:
472:
436:
396:
209:
136:
95:
56:
36:
2462:Stanley, Richard P.
2291:Asymptotic behavior
2266:3464274974065172792
1847:Generating function
671:
529:is a collection of
2596:Integer partitions
2398:
2339:
2305:
2274:
2223:
2197:
2171:
2127:integer partitions
2112:
1919:
1883:
1833:
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1619:
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1492:
1453:
1410:
1367:
1328:
1308:
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1195:
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1145:
1132:
1131:
1098:
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1064:
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1030:
1029:
996:
995:
935:
891:
845:
765:
672:
652:
620:
539:
519:
478:
458:
419:
382:
189:
160:
119:
81:
42:
22:integer partitions
2387:
2324:
2308:{\displaystyle c}
2270:
1646:{\displaystyle 5}
1331:{\displaystyle *}
1171:{\displaystyle 5}
1141:
542:{\displaystyle n}
522:{\displaystyle n}
481:{\displaystyle n}
411:
326:
139:
45:{\displaystyle n}
2603:
2564:
2550:
2543:
2537:
2530:
2524:
2513:
2507:
2505:cond-mat/0303607
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2230:
2229:
2224:
2206:
2204:
2203:
2198:
2180:
2178:
2177:
2172:
2131:plane partitions
2121:
2119:
2118:
2113:
2102:
2101:
2086:
2085:
2070:
2069:
2054:
2053:
2038:
2037:
2010:
2009:
1991:
1990:
1980:
1975:
1948:
1947:
1928:
1926:
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1920:
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1908:
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1884:
1867:
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1411:
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1255:
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1244:
1204:
1202:
1201:
1196:
1194:
1193:
1177:
1175:
1174:
1169:
1154:
1152:
1151:
1146:
1139:
1138:
1134:
1133:
1099:
1065:
1031:
997:
944:
942:
941:
936:
900:
898:
897:
892:
890:
889:
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876:
854:
852:
851:
846:
841:
840:
828:
827:
815:
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786:
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766:
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742:
741:
729:
728:
716:
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700:
681:
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673:
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648:
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487:
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356:
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327:
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321:
320:
296:
295:
264:
263:
239:
238:
198:
196:
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190:
182:
181:
159:
128:
126:
125:
120:
90:
88:
87:
82:
80:
79:
51:
49:
48:
43:
26:plane partitions
18:solid partitions
16:In mathematics,
2611:
2610:
2606:
2605:
2604:
2602:
2601:
2600:
2581:
2580:
2559:
2554:
2553:
2544:
2540:
2531:
2527:
2521:arXiv:1105.6231
2514:
2510:
2498:
2491:
2486:
2482:
2477:
2473:
2460:
2459:
2455:
2447:
2440:
2432:G. E. Andrews,
2431:
2427:
2422:
2418:
2413:
2369:
2349:
2342:
2319:
2318:
2297:
2296:
2293:
2243:
2238:
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2209:
2208:
2183:
2182:
2157:
2156:
2152:
2093:
2077:
2061:
2045:
2029:
2001:
1982:
1939:
1934:
1933:
1900:
1895:
1894:
1858:
1853:
1852:
1849:
1811:
1806:
1805:
1804:with all other
1763:
1738:
1713:
1688:
1663:
1658:
1657:
1635:
1634:
1597:
1584:
1573:
1572:
1548:
1504:
1503:
1470:
1465:
1464:
1431:
1426:
1425:
1388:
1383:
1382:
1381:Given a set of
1345:
1340:
1339:
1320:
1319:
1258:
1257:
1224:
1219:
1218:
1211:
1185:
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1116:
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994:
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987:
986:
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868:
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832:
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746:
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691:
690:
637:
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584:
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410: and
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329:
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243:
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161:
134:
133:
93:
92:
59:
54:
53:
34:
33:
12:
11:
5:
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2558:
2557:External links
2555:
2552:
2551:
2538:
2536:, 158, 950-967
2525:
2519:(2012) 055001
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1560:
1555:
1551:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1489:
1486:
1483:
1480:
1477:
1473:
1450:
1447:
1444:
1441:
1438:
1434:
1407:
1404:
1401:
1398:
1395:
1391:
1379:
1378:
1364:
1361:
1358:
1355:
1352:
1348:
1327:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1243:
1240:
1237:
1234:
1231:
1227:
1210:
1207:
1192:
1188:
1167:
1156:
1155:
1144:
1137:
1128:
1125:
1124:
1121:
1118:
1117:
1114:
1111:
1110:
1107:
1104:
1103:
1094:
1091:
1090:
1087:
1084:
1083:
1080:
1077:
1076:
1073:
1070:
1069:
1060:
1057:
1056:
1053:
1050:
1049:
1046:
1043:
1042:
1039:
1036:
1035:
1026:
1023:
1022:
1019:
1016:
1015:
1012:
1009:
1008:
1005:
1002:
1001:
992:
989:
988:
985:
982:
981:
978:
975:
974:
971:
968:
967:
962:
947:
946:
934:
931:
928:
925:
922:
919:
916:
913:
910:
888:
884:
880:
875:
871:
867:
864:
844:
839:
835:
831:
826:
822:
818:
813:
809:
805:
800:
796:
792:
789:
785:
764:
761:
758:
753:
749:
745:
740:
736:
732:
727:
723:
719:
714:
710:
706:
703:
699:
669:
664:
661:
656:
651:
646:
641:
619:
614:
609:
604:
601:
598:
593:
588:
583:
578:
573:
568:
565:
562:
538:
518:
502:
499:
477:
457:
454:
451:
446:
442:
430:
429:
418:
415:
407:
404:
401:
379:
376:
373:
370:
367:
363:
359:
354:
351:
348:
345:
342:
339:
336:
332:
319:
316:
313:
310:
307:
303:
299:
294:
291:
288:
285:
282:
279:
276:
272:
267:
262:
259:
256:
253:
250:
246:
242:
237:
234:
231:
228:
225:
222:
219:
215:
200:
199:
188:
185:
180:
177:
174:
171:
168:
164:
158:
155:
152:
149:
146:
142:
118:
115:
112:
109:
106:
103:
100:
91:(with indices
78:
75:
72:
69:
66:
62:
41:
13:
10:
9:
6:
4:
3:
2:
2608:
2597:
2594:
2592:
2589:
2588:
2586:
2577:
2574:
2572:
2569:
2567:
2561:
2560:
2556:
2548:
2542:
2539:
2535:
2529:
2526:
2522:
2518:
2512:
2509:
2506:
2502:
2496:
2494:
2490:
2484:
2481:
2475:
2472:
2467:
2463:
2457:
2454:
2450:
2445:
2443:
2439:
2435:
2429:
2426:
2420:
2417:
2410:
2408:
2395:
2392:
2389:
2382:
2378:
2374:
2370:
2362:
2354:
2350:
2346:
2343:
2329:
2316:
2302:
2290:
2288:
2271:
2265:
2262:
2256:
2248:
2244:
2236:
2235:
2234:
2233:. One finds
2220:
2217:
2214:
2194:
2191:
2188:
2168:
2165:
2162:
2149:
2147:
2144:
2140:
2136:
2132:
2128:
2109:
2106:
2103:
2098:
2094:
2090:
2087:
2082:
2078:
2074:
2071:
2066:
2062:
2058:
2055:
2050:
2046:
2042:
2039:
2034:
2030:
2026:
2023:
2020:
2017:
2014:
2011:
2006:
2002:
1995:
1987:
1983:
1972:
1969:
1966:
1962:
1958:
1952:
1944:
1940:
1932:
1931:
1930:
1913:
1905:
1901:
1880:
1877:
1871:
1863:
1859:
1846:
1844:
1828:
1825:
1822:
1819:
1816:
1812:
1788:
1785:
1780:
1777:
1774:
1771:
1768:
1764:
1760:
1755:
1752:
1749:
1746:
1743:
1739:
1735:
1730:
1727:
1724:
1721:
1718:
1714:
1710:
1705:
1702:
1699:
1696:
1693:
1689:
1685:
1680:
1677:
1674:
1671:
1668:
1664:
1656:
1655:
1654:
1640:
1614:
1611:
1608:
1605:
1602:
1598:
1594:
1589:
1585:
1581:
1578:
1553:
1549:
1545:
1542:
1539:
1536:
1533:
1530:
1527:
1524:
1521:
1518:
1515:
1512:
1487:
1484:
1481:
1478:
1475:
1471:
1448:
1445:
1442:
1439:
1436:
1432:
1423:
1422:
1421:
1405:
1402:
1399:
1396:
1393:
1389:
1362:
1359:
1356:
1353:
1350:
1346:
1325:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
1269:
1266:
1241:
1238:
1235:
1232:
1229:
1225:
1216:
1215:
1214:
1208:
1206:
1190:
1186:
1165:
1142:
1135:
1126:
1119:
1112:
1105:
1092:
1085:
1078:
1071:
1058:
1051:
1044:
1037:
1024:
1017:
1010:
1003:
990:
983:
976:
969:
960:
952:
951:
950:
932:
929:
926:
923:
920:
917:
914:
911:
908:
886:
882:
878:
873:
869:
865:
862:
837:
833:
829:
824:
820:
816:
811:
807:
803:
798:
794:
787:
762:
759:
751:
747:
743:
738:
734:
730:
725:
721:
717:
712:
708:
701:
689:If the node
688:
687:Condition FD:
685:
684:
683:
667:
662:
659:
649:
644:
612:
602:
599:
596:
591:
581:
576:
563:
560:
552:
536:
516:
508:
500:
498:
496:
491:
475:
452:
444:
440:
416:
413:
405:
402:
399:
377:
374:
371:
368:
365:
361:
357:
352:
349:
346:
343:
340:
337:
334:
330:
317:
314:
311:
308:
305:
301:
297:
292:
289:
286:
283:
280:
277:
274:
270:
265:
260:
257:
254:
251:
248:
244:
240:
235:
232:
229:
226:
223:
220:
217:
213:
205:
204:
203:
186:
183:
178:
175:
172:
169:
166:
162:
156:
153:
150:
147:
144:
140:
132:
131:
130:
129:) such that
116:
113:
110:
107:
104:
101:
98:
76:
73:
70:
67:
64:
60:
39:
31:
27:
23:
19:
2546:
2541:
2533:
2528:
2516:
2511:
2500:
2483:
2474:
2465:
2456:
2449:
2433:
2428:
2419:
2317:
2294:
2286:
2153:
2124:
1850:
1803:
1632:
1380:
1212:
1157:
948:
686:
550:
504:
489:
431:
201:
17:
15:
2549:(1997) 2281
1843:vanishing.
28:defined by
2585:Categories
2411:References
2315:such that
549:points or
2347:
2336:∞
2333:→
2218:≤
2192:≤
2166:≤
2107:⋯
1978:∞
1963:∑
1878:≡
1582:≤
1540:−
1528:−
1516:−
1326:∗
1303:∗
1294:−
1282:−
1270:−
879:≤
866:≤
763:λ
760:∈
660:≥
650:∈
600:…
561:λ
358:≤
298:≤
241:≤
141:∑
114:≥
2464:(1999).
2143:MacMahon
2139:MacMahon
901:for all
392:for all
630:, with
507:Ferrers
495:Andrews
2269:
1502:nodes
1463:, add
1318:where
1140:
2135:Euler
1929:, by
855:with
551:nodes
2563:OEIS
2137:and
2129:and
1851:Let
1595:<
1571:for
1217:Let
432:Let
202:and
24:and
2344:log
2326:lim
2091:140
325:and
2587::
2547:30
2517:45
2501:36
2492:^
2441:^
2257:72
2221:72
2195:50
2169:28
2075:59
2059:26
2043:10
1959::=
553:,
497:.
2523:.
2396:.
2393:c
2390:=
2383:4
2379:/
2375:3
2371:n
2366:)
2363:n
2360:(
2355:3
2351:p
2330:n
2303:c
2272:,
2263:=
2260:)
2254:(
2249:3
2245:p
2215:n
2189:n
2163:n
2110:.
2104:+
2099:6
2095:q
2088:+
2083:5
2079:q
2072:+
2067:4
2063:q
2056:+
2051:3
2047:q
2040:+
2035:2
2031:q
2027:4
2024:+
2021:q
2018:+
2015:1
2012:=
2007:n
2003:q
1999:)
1996:n
1993:(
1988:3
1984:p
1973:0
1970:=
1967:n
1956:)
1953:q
1950:(
1945:3
1941:P
1917:)
1914:q
1911:(
1906:3
1902:P
1881:1
1875:)
1872:0
1869:(
1864:3
1860:p
1829:k
1826:,
1823:j
1820:,
1817:i
1813:n
1789:1
1786:=
1781:1
1778:,
1775:2
1772:,
1769:2
1765:n
1761:=
1756:2
1753:,
1750:1
1747:,
1744:1
1740:n
1736:=
1731:1
1728:,
1725:2
1722:,
1719:1
1715:n
1711:=
1706:1
1703:,
1700:1
1697:,
1694:2
1690:n
1686:=
1681:1
1678:,
1675:1
1672:,
1669:1
1665:n
1641:5
1615:k
1612:,
1609:j
1606:,
1603:i
1599:n
1590:4
1586:y
1579:0
1559:)
1554:4
1550:y
1546:,
1543:1
1537:k
1534:,
1531:1
1525:j
1522:,
1519:1
1513:i
1510:(
1488:k
1485:,
1482:j
1479:,
1476:i
1472:n
1449:k
1446:,
1443:j
1440:,
1437:i
1433:n
1406:k
1403:,
1400:j
1397:,
1394:i
1390:n
1363:k
1360:,
1357:j
1354:,
1351:i
1347:n
1306:)
1300:,
1297:1
1291:k
1288:,
1285:1
1279:j
1276:,
1273:1
1267:i
1264:(
1242:k
1239:,
1236:j
1233:,
1230:i
1226:n
1191:4
1187:S
1166:5
1143:,
1136:)
1127:0
1120:0
1113:1
1106:1
1093:0
1086:0
1079:0
1072:1
1059:0
1052:0
1045:1
1038:0
1025:0
1018:1
1011:0
1004:0
991:0
984:0
977:0
970:0
961:(
945:.
933:4
930:,
927:3
924:,
921:2
918:,
915:1
912:=
909:i
887:i
883:a
874:i
870:y
863:0
843:)
838:4
834:y
830:,
825:3
821:y
817:,
812:2
808:y
804:,
799:1
795:y
791:(
788:=
784:y
757:)
752:4
748:a
744:,
739:3
735:a
731:,
726:2
722:a
718:,
713:1
709:a
705:(
702:=
698:a
668:4
663:0
655:Z
645:i
640:y
618:)
613:n
608:y
603:,
597:,
592:2
587:y
582:,
577:1
572:y
567:(
564:=
537:n
517:n
476:n
456:)
453:n
450:(
445:3
441:p
417:.
414:k
406:j
403:,
400:i
378:k
375:,
372:j
369:,
366:i
362:n
353:1
350:+
347:k
344:,
341:j
338:,
335:i
331:n
318:k
315:,
312:j
309:,
306:i
302:n
293:k
290:,
287:1
284:+
281:j
278:,
275:i
271:n
266:,
261:k
258:,
255:j
252:,
249:i
245:n
236:k
233:,
230:j
227:,
224:1
221:+
218:i
214:n
187:n
184:=
179:k
176:,
173:j
170:,
167:i
163:n
157:k
154:,
151:j
148:,
145:i
117:1
111:k
108:,
105:j
102:,
99:i
77:k
74:,
71:j
68:,
65:i
61:n
40:n
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