28:
837:
623:
Whether these guesses are warranted by evidence, I leave it to the reader to decide. Whatever the final verdict of posterity may be, I believe the "crank" is unique among arithmetical functions in having been named before it was discovered. May it be preserved from the ignominious fate of the planet
2274:
and his coauthors argued that
Ramanujan knew about the crank, although not in the form that Andrews and Garvan have defined. In a systematic study of the Lost Notebook of Ramanujan, Berndt and his coauthors have given substantial evidence that Ramanujan knew about the dissections of the crank
1776:
153:+ 6 ) can be divided into 5 (respectively, 7 and 11) subclasses of equal size. The then known proofs of these congruences were based on the ideas of generating functions and they did not specify a method for the division of the partitions into subclasses of equal size.
702:
1965:
The concepts of rank and crank can both be used to classify partitions of certain integers into subclasses of equal size. However the two concepts produce different subclasses of partitions. This is illustrated in the following two tables.
160:. The rank of a partition is the integer obtained by subtracting the number of parts in the partition from the largest part in the partition. For example, the rank of the partition λ = { 4, 2, 1, 1, 1 } of 9 is 4 − 5 = −1. Denoting by
1567:
340:+ 4 into five classes of equal size: Put in one class all those partitions whose ranks are congruent to each other modulo 5. The same idea can be applied to divide the partitions of integers of the form 7
494:
that there exists an arithmetical coefficient similar to, but more recondite than, the rank of a partition; I shall call this hypothetical coefficient the "crank" of the partition and denote by
832:{\displaystyle c(\lambda )={\begin{cases}\ell (\lambda )&{\text{if }}\omega (\lambda )=0\\\mu (\lambda )-\omega (\lambda )&{\text{if }}\omega (\lambda )>0.\end{cases}}}
2493:
1771:{\displaystyle \sum _{n=0}^{\infty }\sum _{m=-\infty }^{\infty }M(m,n)z^{m}q^{n}=\prod _{n=1}^{\infty }{\frac {(1-q^{n})}{(1-zq^{n})(1-z^{-1}q^{n})}}}
51:, who hypothesised its existence in a 1944 paper. Dyson gave a list of properties this yet-to-be-defined quantity should have. In 1988,
92:
2312:
1786:
Andrews and Garvan proved the following result which shows that the crank as defined above does meet the conditions given by Dyson.
2334:
2483:
2488:
625:
2478:
336:
Assuming that these conjectures are true, they provided a way of splitting up all partitions of numbers of the form 5
637:
In a paper published in 1988 George E. Andrews and F. G. Garvan defined the crank of a partition as follows:
344:+ 5 into seven equally numerous classes. But the idea fails to divide partitions of integers of the form 11
103:
2454:
2417:
726:
47:
is a certain number associated with the partition. It was first introduced without a definition by
157:
88:
2444:
2407:
2308:
59:
discovered a definition for the crank satisfying the properties hypothesized for it by Dyson.
52:
44:
844:
The cranks of the partitions of the integers 4, 5, 6 are computed in the following tables.
2271:
484:
Thus the rank cannot be used to prove the theorem combinatorially. However, Dyson wrote,
2458:
2421:
2293:
2435:
Manjil P. Saikia (2015). "A study of the crank function in
Ramanujan's Lost Notebook".
2472:
1518:(1,1) = 1 as given by the following generating function. The number of partitions of
48:
36:
220:. Based on empirical evidences Dyson formulated the following conjectures known as
56:
17:
91:
in a paper published in 1918 stated and proved the following congruences for the
27:
348:+ 6 into 11 classes of the same size, as the following table shows.
2449:
2412:
26:
1970:
Classification of the partitions of the integer 9 based on cranks
2398:
Manjil P. Saikia (2013). "Cranks in
Ramanujan's Lost Notebook".
2118:
Classification of the partitions of the integer 9 based on ranks
141:
These congruences imply that partitions of numbers of the form 5
825:
156:
In his Eureka paper Dyson proposed the concept of the
1570:
705:
2381:Proceedings of the Cambridge Philosophical Society
1770:
831:
2367:Srinivasa, Ramanujan (1919). "Some properties of
2342:Bulletin of the American Mathematical Society
2333:George E. Andrews; F.G. Garvan (April 1988).
8:
2400:Journal of the Assam Academy of Mathematics
2294:"Some Guesses in The Theory of Partitions"
2448:
2411:
1756:
1743:
1721:
1694:
1678:
1672:
1661:
1648:
1638:
1610:
1596:
1586:
1575:
1569:
799:
743:
721:
704:
360:= 0 ) divided into classes based on ranks
2121:
1973:
1207:
1012:
851:
363:
2284:
2328:
2326:
2324:
79:) denote the number of partitions of
7:
1673:
1611:
1606:
1587:
71:be a non-negative integer and let
25:
2494:Conjectures that have been proved
145:+ 4 (respectively, of the forms 7
677:) denote the number of parts of
352:Partitions of the integer 6 ( 11
665:) denote the number of 1's in
176:), the number of partitions of
2335:"Dyson's crank of a partition"
1762:
1730:
1727:
1705:
1700:
1681:
1631:
1619:
1482:, the number of partitions of
813:
807:
794:
788:
779:
773:
757:
751:
738:
732:
715:
709:
227:For all non-negative integers
1:
2253:{ 1, 1, 1, 1, 1, 1, 1, 1, 1 }
2102:{ 1, 1, 1, 1, 1, 1, 1, 1, 1 }
1204:Cranks of the partitions of 6
1009:Cranks of the partitions of 5
848:Cranks of the partitions of 4
653:) denote the largest part of
180:whose ranks are congruent to
1549:The generating function for
2375:), number of partitions of
2259:{ 2, 1, 1, 1, 1, 1, 1, 1 }
2246:
2238:
2204:
2201:
2195:
2164:
2098:
2081:
2064:
2047:
2030:
2013:
519:whose crank is congruent to
513:the number of partitions of
212:+ 5) for various values of
2510:
2108:{ 2, 1, 1, 1, 1, 1, 1, 1}
2292:Freeman J. Dyson (1944).
87:(0) is defined to be 1).
2111:{ 3, 1, 1, 1, 1, 1, 1 }
2437:The Mathematics Student
2247:{ 2, 2, 1, 1, 1, 1, 1 }
2233:{ 3, 1, 1, 1, 1, 1, 1 }
2099:{ 2, 2, 1, 1, 1, 1, 1 }
1772:
1677:
1615:
1591:
833:
32:
2275:generating function.
1773:
1657:
1592:
1571:
1506: = 1 where
1478:≥ 0 and all integers
834:
488:I hold in fact :
104:Ramanujan congruences
62:
31:Freeman Dyson in 2005
30:
2484:Arithmetic functions
2266:Ramanujan and cranks
2250:{ 2, 2, 2, 1, 1, 1 }
2239:{ 4, 1, 1, 1, 1, 1 }
2219:{ 3, 2, 1, 1, 1, 1 }
2105:{ 2, 2, 2, 1, 1, 1 }
2085:{ 3, 2, 1, 1, 1, 1 }
2082:{ 4, 1, 1, 1, 1, 1 }
1568:
1522:with crank equal to
1486:with crank equal to
703:
2489:Srinivasa Ramanujan
2459:2014arXiv1406.3299S
2422:2014arXiv1402.6644S
633:Definition of crank
188:, Dyson considered
158:rank of a partition
89:Srinivasa Ramanujan
18:Crank (mathematics)
2479:Integer partitions
2301:Eureka (Cambridge)
2208:{ 4, 2, 1, 1, 1 }
2094:{ 3, 2, 2, 1, 1 }
1768:
1561:) is given below:
829:
824:
102:), since known as
93:partition function
33:
2263:
2262:
2236:{ 2, 2, 2, 2, 1 }
2222:{ 3, 3, 1, 1, 1 }
2179:{ 5, 1, 1, 1, 1 }
2115:
2114:
2091:{ 2, 2, 2, 2, 1 }
2088:{ 3, 3, 1, 1, 1 }
2040:{ 4, 2, 1, 1, 1 }
2037:{ 5, 1, 1, 1, 1 }
1766:
1474:For all integers
1464:
1463:
1201:
1200:
1006:
1005:
802:
746:
482:
481:
137:+ 6) ≡ 0 (mod 11)
53:George E. Andrews
45:integer partition
16:(Redirected from
2501:
2463:
2462:
2452:
2432:
2426:
2425:
2415:
2395:
2389:
2388:
2364:
2358:
2357:
2355:
2353:
2339:
2330:
2319:
2318:
2298:
2289:
2256:{ 3, 2, 2, 1, 1}
2153:Partitions with
2146:Partitions with
2139:Partitions with
2132:Partitions with
2125:Partitions with
2122:
2007:crank ≡ 4
2005:Partitions with
1998:Partitions with
1991:Partitions with
1984:Partitions with
1977:Partitions with
1974:
1777:
1775:
1774:
1769:
1767:
1765:
1761:
1760:
1751:
1750:
1726:
1725:
1703:
1699:
1698:
1679:
1676:
1671:
1653:
1652:
1643:
1642:
1614:
1609:
1590:
1585:
1208:
1013:
852:
838:
836:
835:
830:
828:
827:
803:
800:
747:
744:
641:For a partition
364:
222:rank conjectures
127:+ 5) ≡ 0 (mod 7)
117:+ 4) ≡ 0 (mod 5)
21:
2509:
2508:
2504:
2503:
2502:
2500:
2499:
2498:
2469:
2468:
2467:
2466:
2434:
2433:
2429:
2397:
2396:
2392:
2366:
2365:
2361:
2351:
2349:
2337:
2332:
2331:
2322:
2315:
2296:
2291:
2290:
2286:
2281:
2272:Bruce C. Berndt
2270:Recent work by
2268:
2242:{ 3, 2, 2, 2 }
2225:{ 3, 3, 2, 1 }
2156:
2155:rank ≡ 4
2154:
2149:
2148:rank ≡ 3
2147:
2142:
2141:rank ≡ 2
2140:
2135:
2133:
2128:
2127:rank ≡ 0
2126:
2120:
2060:{ 5, 2, 1, 1 }
2008:
2006:
2001:
1999:
1994:
1992:
1987:
1985:
1980:
1978:
1972:
1937:+ 6) = . . . =
1784:
1752:
1739:
1717:
1704:
1690:
1680:
1644:
1634:
1566:
1565:
1472:
1466:
1265:
1253:
1243:
1242:Number of parts
1231:
1219:
1212:
1206:
1070:
1058:
1048:
1047:Number of parts
1036:
1024:
1017:
1011:
909:
897:
887:
886:Number of parts
875:
863:
856:
850:
823:
822:
797:
767:
766:
741:
722:
701:
700:
635:
418:
413:
408:
403:
398:
393:
388:
383:
378:
373:
368:
362:
65:
23:
22:
15:
12:
11:
5:
2507:
2505:
2497:
2496:
2491:
2486:
2481:
2471:
2470:
2465:
2464:
2427:
2390:
2359:
2344:. New Series.
2320:
2313:
2283:
2282:
2280:
2277:
2267:
2264:
2261:
2260:
2257:
2254:
2251:
2248:
2244:
2243:
2240:
2237:
2234:
2231:
2227:
2226:
2223:
2220:
2217:
2214:
2213:{ 4, 2, 2, 1 }
2210:
2209:
2206:
2203:
2200:
2197:
2196:{ 4, 3, 1, 1 }
2193:
2192:
2189:
2186:
2183:
2182:{ 5, 2, 1, 1 }
2180:
2176:
2175:
2172:
2169:
2168:{ 6, 1, 1, 1 }
2166:
2163:
2159:
2158:
2151:
2144:
2137:
2134:rank ≡ 1
2130:
2116:
2113:
2112:
2109:
2106:
2103:
2100:
2096:
2095:
2092:
2089:
2086:
2083:
2079:
2078:
2075:
2074:{ 3, 2, 2, 2 }
2072:
2071:{ 3, 3, 2, 1 }
2069:
2066:
2065:{ 4, 3, 1, 1 }
2062:
2061:
2058:
2055:
2054:{ 4, 2, 2, 1 }
2052:
2049:
2045:
2044:
2041:
2038:
2035:
2032:
2028:
2027:
2024:
2023:{ 6, 1, 1, 1 }
2021:
2018:
2015:
2011:
2010:
2003:
1996:
1989:
1982:
1968:
1963:
1962:
1904:
1838:
1783:
1780:
1779:
1778:
1764:
1759:
1755:
1749:
1746:
1742:
1738:
1735:
1732:
1729:
1724:
1720:
1716:
1713:
1710:
1707:
1702:
1697:
1693:
1689:
1686:
1683:
1675:
1670:
1667:
1664:
1660:
1656:
1651:
1647:
1641:
1637:
1633:
1630:
1627:
1624:
1621:
1618:
1613:
1608:
1605:
1602:
1599:
1595:
1589:
1584:
1581:
1578:
1574:
1530:is denoted by
1490:is denoted by
1471:
1468:
1462:
1461:
1458:
1455:
1452:
1449:
1445:
1444:
1441:
1438:
1435:
1432:
1428:
1427:
1424:
1421:
1418:
1415:
1411:
1410:
1407:
1404:
1401:
1398:
1394:
1393:
1390:
1387:
1384:
1381:
1377:
1376:
1373:
1370:
1367:
1364:
1360:
1359:
1356:
1353:
1350:
1347:
1343:
1342:
1339:
1336:
1333:
1330:
1326:
1325:
1322:
1319:
1316:
1313:
1309:
1308:
1305:
1302:
1299:
1296:
1292:
1291:
1288:
1285:
1282:
1279:
1275:
1274:
1262:
1240:
1230:Number of 1's
1228:
1216:
1202:
1199:
1198:
1195:
1192:
1189:
1186:
1182:
1181:
1178:
1175:
1172:
1169:
1165:
1164:
1161:
1158:
1155:
1152:
1148:
1147:
1144:
1141:
1138:
1135:
1131:
1130:
1127:
1124:
1121:
1118:
1114:
1113:
1110:
1107:
1104:
1101:
1097:
1096:
1093:
1090:
1087:
1084:
1080:
1079:
1067:
1045:
1035:Number of 1's
1033:
1021:
1007:
1004:
1003:
1000:
997:
994:
991:
987:
986:
983:
980:
977:
974:
970:
969:
966:
963:
960:
957:
953:
952:
949:
946:
943:
940:
936:
935:
932:
929:
926:
923:
919:
918:
906:
884:
874:Number of 1's
872:
860:
846:
842:
841:
840:
839:
826:
821:
818:
815:
812:
809:
806:
798:
796:
793:
790:
787:
784:
781:
778:
775:
772:
769:
768:
765:
762:
759:
756:
753:
750:
742:
740:
737:
734:
731:
728:
727:
725:
720:
717:
714:
711:
708:
697:) is given by
634:
631:
620:
619:
614:
569:
528:
480:
479:
476:
474:
472:
470:
468:
466:
464:
462:
460:
457:
454:
453:
450:
447:
444:
442:
439:
436:
434:
431:
428:
425:
421:
420:
415:
410:
405:
400:
395:
390:
385:
380:
375:
370:
350:
334:
333:
275:
139:
138:
128:
118:
64:
61:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2506:
2495:
2492:
2490:
2487:
2485:
2482:
2480:
2477:
2476:
2474:
2460:
2456:
2451:
2446:
2442:
2438:
2431:
2428:
2423:
2419:
2414:
2409:
2405:
2401:
2394:
2391:
2386:
2382:
2378:
2374:
2370:
2363:
2360:
2347:
2343:
2336:
2329:
2327:
2325:
2321:
2316:
2314:9780821805619
2310:
2306:
2302:
2295:
2288:
2285:
2278:
2276:
2273:
2265:
2258:
2255:
2252:
2249:
2245:
2241:
2235:
2232:
2229:
2228:
2224:
2221:
2218:
2215:
2212:
2211:
2207:
2198:
2194:
2190:
2187:
2184:
2181:
2178:
2177:
2173:
2170:
2167:
2161:
2160:
2152:
2145:
2138:
2131:
2124:
2123:
2119:
2110:
2107:
2104:
2101:
2097:
2093:
2090:
2087:
2084:
2080:
2076:
2073:
2070:
2067:
2063:
2059:
2056:
2053:
2050:
2046:
2042:
2039:
2036:
2033:
2029:
2025:
2022:
2019:
2016:
2012:
2004:
1997:
1990:
1983:
1976:
1975:
1971:
1967:
1960:
1956:
1952:
1948:
1944:
1940:
1936:
1932:
1928:
1924:
1920:
1916:
1912:
1908:
1905:
1902:
1898:
1894:
1890:
1886:
1882:
1878:
1874:
1870:
1866:
1862:
1858:
1854:
1850:
1846:
1842:
1839:
1836:
1832:
1828:
1824:
1820:
1816:
1812:
1808:
1804:
1800:
1796:
1792:
1789:
1788:
1787:
1781:
1757:
1753:
1747:
1744:
1740:
1736:
1733:
1722:
1718:
1714:
1711:
1708:
1695:
1691:
1687:
1684:
1668:
1665:
1662:
1658:
1654:
1649:
1645:
1639:
1635:
1628:
1625:
1622:
1616:
1603:
1600:
1597:
1593:
1582:
1579:
1576:
1572:
1564:
1563:
1562:
1560:
1556:
1552:
1547:
1545:
1541:
1537:
1533:
1529:
1525:
1521:
1517:
1513:
1509:
1505:
1502:) except for
1501:
1497:
1493:
1489:
1485:
1481:
1477:
1469:
1467:
1459:
1456:
1453:
1450:
1448:{1,1,1,1,1,1}
1447:
1446:
1442:
1439:
1436:
1433:
1430:
1429:
1425:
1422:
1419:
1416:
1413:
1412:
1408:
1405:
1402:
1399:
1396:
1395:
1391:
1388:
1385:
1382:
1379:
1378:
1374:
1371:
1368:
1365:
1362:
1361:
1357:
1354:
1351:
1348:
1345:
1344:
1340:
1337:
1334:
1331:
1328:
1327:
1323:
1320:
1317:
1314:
1311:
1310:
1306:
1303:
1300:
1297:
1294:
1293:
1289:
1286:
1283:
1280:
1277:
1276:
1272:
1268:
1263:
1260:
1256:
1251:
1247:
1241:
1238:
1234:
1229:
1226:
1222:
1218:Largest part
1217:
1215:
1210:
1209:
1205:
1196:
1193:
1190:
1187:
1184:
1183:
1179:
1176:
1173:
1170:
1167:
1166:
1162:
1159:
1156:
1153:
1150:
1149:
1145:
1142:
1139:
1136:
1133:
1132:
1128:
1125:
1122:
1119:
1116:
1115:
1111:
1108:
1105:
1102:
1099:
1098:
1094:
1091:
1088:
1085:
1082:
1081:
1077:
1073:
1068:
1065:
1061:
1056:
1052:
1046:
1043:
1039:
1034:
1031:
1027:
1023:Largest part
1022:
1020:
1015:
1014:
1010:
1001:
998:
995:
992:
989:
988:
984:
981:
978:
975:
972:
971:
967:
964:
961:
958:
955:
954:
950:
947:
944:
941:
938:
937:
933:
930:
927:
924:
921:
920:
916:
912:
907:
904:
900:
895:
891:
885:
882:
878:
873:
870:
866:
862:Largest part
861:
859:
854:
853:
849:
845:
819:
816:
810:
804:
791:
785:
782:
776:
770:
763:
760:
754:
748:
735:
729:
723:
718:
712:
706:
699:
698:
696:
692:
689:). The crank
688:
684:
680:
676:
672:
668:
664:
660:
656:
652:
648:
644:
640:
639:
638:
632:
630:
629:
627:
618:
615:
612:
608:
604:
600:
596:
592:
588:
584:
580:
576:
573:
570:
567:
563:
559:
555:
551:
547:
543:
539:
535:
532:
529:
527:
523:
520:
517:
514:
510:
506:
502:
498:
495:
492:
491:
490:
489:
485:
477:
475:
473:
471:
469:
467:
465:
463:
461:
458:
456:
455:
451:
448:
445:
443:
441:{1,1,1,1,1,1}
440:
437:
435:
432:
429:
426:
423:
422:
416:
411:
406:
401:
396:
391:
386:
381:
376:
371:
366:
365:
361:
359:
355:
349:
347:
343:
339:
331:
327:
323:
319:
315:
311:
307:
303:
299:
295:
291:
287:
283:
279:
276:
273:
269:
265:
261:
257:
253:
249:
245:
241:
237:
234:
233:
232:
230:
225:
223:
219:
215:
211:
207:
203:
199:
195:
191:
187:
183:
179:
175:
171:
167:
163:
159:
154:
152:
148:
144:
136:
132:
129:
126:
122:
119:
116:
112:
109:
108:
107:
105:
101:
97:
94:
90:
86:
82:
78:
74:
70:
63:Dyson's crank
60:
58:
54:
50:
49:Freeman Dyson
46:
42:
38:
37:number theory
29:
19:
2440:
2436:
2430:
2403:
2399:
2393:
2384:
2380:
2376:
2372:
2368:
2362:
2350:. Retrieved
2345:
2341:
2304:
2300:
2287:
2269:
2174:{ 7, 1, 1 }
2117:
2077:{ 4, 3, 2 }
2043:{ 7, 1, 1 }
1969:
1964:
1958:
1954:
1950:
1946:
1942:
1938:
1934:
1930:
1926:
1922:
1918:
1914:
1910:
1906:
1900:
1896:
1892:
1888:
1884:
1880:
1876:
1872:
1868:
1864:
1860:
1856:
1852:
1848:
1844:
1840:
1834:
1830:
1826:
1822:
1818:
1814:
1810:
1806:
1802:
1798:
1794:
1790:
1785:
1782:Basic result
1558:
1554:
1550:
1548:
1543:
1539:
1535:
1531:
1527:
1523:
1519:
1515:
1511:
1507:
1503:
1499:
1495:
1491:
1487:
1483:
1479:
1475:
1473:
1465:
1270:
1266:
1258:
1254:
1249:
1245:
1244:larger than
1236:
1232:
1224:
1220:
1213:
1203:
1075:
1071:
1063:
1059:
1054:
1050:
1049:larger than
1041:
1037:
1029:
1025:
1018:
1008:
914:
910:
902:
898:
893:
889:
888:larger than
880:
876:
868:
864:
857:
847:
843:
694:
690:
686:
682:
681:larger than
678:
674:
670:
666:
662:
658:
654:
650:
646:
642:
636:
622:
621:
616:
610:
606:
602:
598:
594:
590:
586:
582:
578:
574:
571:
565:
561:
557:
553:
549:
545:
541:
537:
533:
530:
525:
521:
518:
515:
512:
508:
504:
500:
496:
493:
487:
486:
483:
357:
353:
351:
345:
341:
337:
335:
329:
325:
321:
317:
313:
309:
305:
301:
297:
293:
289:
285:
281:
277:
271:
267:
263:
259:
255:
251:
247:
243:
239:
235:
228:
226:
221:
217:
213:
209:
205:
201:
197:
193:
189:
185:
181:
177:
173:
169:
165:
161:
155:
150:
146:
142:
140:
134:
130:
124:
120:
114:
110:
99:
95:
84:
80:
76:
72:
68:
66:
57:Frank Garvan
40:
34:
2352:26 November
2230:{ 3, 3, 3 }
2216:{ 4, 3, 2 }
2202:{ 5, 2, 2 }
2199:{ 4, 4, 1 }
2188:{ 6, 2, 1 }
2068:{ 4, 4, 1 }
2057:{ 3, 3, 3 }
2051:{ 5, 3, 1 }
2048:{ 5, 2, 2 }
2034:{ 6, 2, 1 }
1949:(10, 11, 11
1431:{2,1,1,1,1}
1185:{1,1,1,1,1}
617:that . . .
446:{2,1,1,1,1}
2473:Categories
2387:: 207–210.
2279:References
2185:{ 5, 3, 1}
2000:crank ≡ 3
1993:crank ≡ 2
1979:crank ≡ 0
1941:(9, 11, 11
1933:(3, 11, 11
1925:(2, 11, 11
1917:(1, 11, 11
1909:(0, 11, 11
1510:(−1,1) = −
1211:Partition
1016:Partition
855:Partition
609:(4, 11, 11
601:(3, 11, 11
593:(2, 11, 11
585:(1, 11, 11
577:(0, 11, 11
478:{3,1,1,1}
417:rank ≡ 10
149:+ 5 and 11
2450:1406.3299
2413:1402.6644
2307:: 10–15.
2191:{ 6, 3 }
2017:{ 6, 3 }
1986:crank ≡ 1
1961:+ 6) / 11
1745:−
1737:−
1712:−
1688:−
1674:∞
1659:∏
1612:∞
1607:∞
1604:−
1594:∑
1588:∞
1573:∑
1470:Notations
1414:{2,2,1,1}
1380:{3,1,1,1}
1168:{2,1,1,1}
990:{1,1,1,1}
811:λ
805:ω
792:λ
786:ω
783:−
777:λ
771:μ
755:λ
749:ω
736:λ
730:ℓ
713:λ
449:{2,2,1,1}
419:(mod 11)
412:rank ≡ 9
407:rank ≡ 8
402:rank ≡ 7
392:rank ≡ 5
387:rank ≡ 4
382:rank ≡ 3
377:rank ≡ 2
372:rank ≡ 1
367:rank ≡ 0
356:+ 6 with
231:we have:
200:+ 4) and
2205:{ 5, 4 }
2165:{ 8, 1 }
2162:{ 7, 2 }
2157:(mod 5)
2031:{ 5, 4 }
2020:{ 7, 2 }
2014:{ 8, 1 }
2009:(mod 5)
1903:+ 5) / 7
1891:(6, 7, 7
1883:(5, 7, 7
1875:(4, 7, 7
1867:(3, 7, 7
1859:(2, 7, 7
1851:(1, 7, 7
1843:(0, 7, 7
1837:+ 4) / 5
1825:(4, 5, 5
1817:(3, 5, 5
1809:(2, 5, 5
1801:(1, 5, 5
1793:(0, 5, 5
1514:(0,1) =
801:if
745:if
452:{2,2,2}
414:(mod 11)
409:(mod 11)
404:(mod 11)
399:(mod 11)
397:rank ≡ 6
394:(mod 11)
389:(mod 11)
384:(mod 11)
379:(mod 11)
374:(mod 11)
369:(mod 11)
328:(6, 7, 7
320:(5, 7, 7
312:(4, 7, 7
304:(3, 7, 7
296:(2, 7, 7
288:(1, 7, 7
280:(0, 7, 7
270:(4, 5, 5
262:(3, 5, 5
258:+ 4) =
254:(2, 5, 5
250:+ 4) =
246:(1, 5, 5
242:+ 4) =
238:(0, 5, 5
2455:Bibcode
2418:Bibcode
2150:(mod 5)
2143:(mod 5)
2136:(mod 5)
2129:(mod 5)
2002:(mod 5)
1995:(mod 5)
1988:(mod 5)
1981:(mod 5)
1953:+ 6) =
1945:+ 6) =
1929:+ 6) =
1921:+ 6) =
1913:+ 6) =
1895:+ 5) =
1887:+ 5) =
1879:+ 5) =
1871:+ 5) =
1863:+ 5) =
1855:+ 5) =
1847:+ 5) =
1829:+ 4) =
1821:+ 4) =
1813:+ 4) =
1805:+ 4) =
1797:+ 4) =
1526:modulo
1397:{2,2,2}
1363:{3,2,1}
1329:{4,1,1}
1151:{2,2,1}
1134:{3,1,1}
973:{2,1,1}
605:+ 6) =
597:+ 6) =
589:+ 6) =
581:+ 6) =
524:modulo
427:{4,1,1}
424:{3,2,1}
324:+ 5) =
316:+ 5) =
308:+ 5) =
300:+ 5) =
292:+ 5) =
284:+ 5) =
266:+ 4) =
196:, 5, 5
184:modulo
2311:
2026:{ 9 }
669:, and
645:, let
626:Vulcan
208:, 7, 7
43:of an
39:, the
2445:arXiv
2408:arXiv
2338:(PDF)
2297:(PDF)
2171:{ 9 }
1346:{3,3}
1312:{4,2}
1295:{5,1}
1264:Crank
1117:{3,2}
1100:{4,1}
1069:Crank
956:{2,2}
939:{3,1}
908:Crank
613:+ 6);
459:{3,3}
433:{5,1}
430:{4,2}
274:+ 4).
41:crank
2354:2012
2309:ISBN
817:>
572:that
548:) =
531:that
332:+ 5)
216:and
67:Let
55:and
2385:XIX
2379:".
2348:(2)
1957:(11
1546:).
1460:−6
1443:−4
1426:−2
1392:−3
1341:−1
1278:{6}
1197:−5
1180:−3
1146:−1
1083:{5}
1002:−4
985:−2
922:{4}
438:{6}
133:(11
35:In
2475::
2453:.
2443:.
2441:84
2439:.
2416:.
2406:.
2402:.
2383:.
2346:18
2340:.
2323:^
2303:.
2299:.
1899:(7
1833:(5
1409:2
1375:1
1358:3
1324:4
1307:0
1290:6
1273:)
1163:1
1129:3
1112:0
1095:5
1078:)
968:2
951:0
934:4
917:)
820:0.
657:,
628:.
568:);
564:,
560:,
556:−
544:,
540:,
526:q;
511:)
507:,
503:,
224:.
172:,
168:,
123:(7
113:(5
106:.
2461:.
2457::
2447::
2424:.
2420::
2410::
2404:6
2377:n
2373:n
2371:(
2369:p
2356:.
2317:.
2305:8
1959:n
1955:p
1951:n
1947:M
1943:n
1939:M
1935:n
1931:M
1927:n
1923:M
1919:n
1915:M
1911:n
1907:M
1901:n
1897:p
1893:n
1889:M
1885:n
1881:M
1877:n
1873:M
1869:n
1865:M
1861:n
1857:M
1853:n
1849:M
1845:n
1841:M
1835:n
1831:p
1827:n
1823:M
1819:n
1815:M
1811:n
1807:M
1803:n
1799:M
1795:n
1791:M
1763:)
1758:n
1754:q
1748:1
1741:z
1734:1
1731:(
1728:)
1723:n
1719:q
1715:z
1709:1
1706:(
1701:)
1696:n
1692:q
1685:1
1682:(
1669:1
1666:=
1663:n
1655:=
1650:n
1646:q
1640:m
1636:z
1632:)
1629:n
1626:,
1623:m
1620:(
1617:M
1601:=
1598:m
1583:0
1580:=
1577:n
1559:n
1557:,
1555:m
1553:(
1551:M
1544:n
1542:,
1540:q
1538:,
1536:m
1534:(
1532:M
1528:q
1524:m
1520:n
1516:M
1512:M
1508:M
1504:n
1500:n
1498:,
1496:m
1494:(
1492:M
1488:m
1484:n
1480:m
1476:n
1457:0
1454:6
1451:1
1440:0
1437:4
1434:2
1423:0
1420:2
1417:2
1406:3
1403:0
1400:2
1389:0
1386:3
1383:3
1372:2
1369:1
1366:3
1355:2
1352:0
1349:3
1338:1
1335:2
1332:4
1321:2
1318:0
1315:4
1304:1
1301:1
1298:5
1287:1
1284:0
1281:6
1271:λ
1269:(
1267:c
1261:)
1259:λ
1257:(
1255:μ
1252:)
1250:λ
1248:(
1246:ω
1239:)
1237:λ
1235:(
1233:ω
1227:)
1225:λ
1223:(
1221:ℓ
1214:λ
1194:0
1191:5
1188:1
1177:0
1174:3
1171:2
1160:2
1157:1
1154:2
1143:1
1140:2
1137:3
1126:2
1123:0
1120:3
1109:1
1106:1
1103:4
1092:1
1089:0
1086:5
1076:λ
1074:(
1072:c
1066:)
1064:λ
1062:(
1060:μ
1057:)
1055:λ
1053:(
1051:ω
1044:)
1042:λ
1040:(
1038:ω
1032:)
1030:λ
1028:(
1026:ℓ
1019:λ
999:0
996:4
993:1
982:0
979:2
976:2
965:2
962:0
959:2
948:1
945:1
942:3
931:1
928:0
925:4
915:λ
913:(
911:c
905:)
903:λ
901:(
899:μ
896:)
894:λ
892:(
890:ω
883:)
881:λ
879:(
877:ω
871:)
869:λ
867:(
865:ℓ
858:λ
814:)
808:(
795:)
789:(
780:)
774:(
764:0
761:=
758:)
752:(
739:)
733:(
724:{
719:=
716:)
710:(
707:c
695:λ
693:(
691:c
687:λ
685:(
683:ω
679:λ
675:λ
673:(
671:μ
667:λ
663:λ
661:(
659:ω
655:λ
651:λ
649:(
647:ℓ
643:λ
611:n
607:M
603:n
599:M
595:n
591:M
587:n
583:M
579:n
575:M
566:n
562:q
558:m
554:q
552:(
550:M
546:n
542:q
538:m
536:(
534:M
522:m
516:n
509:n
505:q
501:m
499:(
497:M
358:n
354:n
346:n
342:n
338:n
330:n
326:N
322:n
318:N
314:n
310:N
306:n
302:N
298:n
294:N
290:n
286:N
282:n
278:N
272:n
268:N
264:n
260:N
256:n
252:N
248:n
244:N
240:n
236:N
229:n
218:m
214:n
210:n
206:m
204:(
202:N
198:n
194:m
192:(
190:N
186:q
182:m
178:n
174:n
170:q
166:m
164:(
162:N
151:n
147:n
143:n
135:n
131:p
125:n
121:p
115:n
111:p
100:n
98:(
96:p
85:p
83:(
81:n
77:n
75:(
73:p
69:n
20:)
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