20:
1597:
282:
817:
1438: = 6, the partitions and have associated sequences (0,3,4,5,6,6,6) and (0,2,4,6,6,6,6), whose componentwise maximum (0,3,4,6,6,6,6) does not correspond to any partition. To show that any two partitions of
1095:
1425:
1328:
1527:
624:
181:
912:
1251:
995:
1174:
581:
96:
688:
1330:
The componentwise minimum of two nondecreasing concave integer sequences is also nondecreasing and concave. Therefore, for any two partitions of
1002:
1357:
1256:
1464:
1775:
1585:
1726:
1702:
1780:
277:{\displaystyle p\trianglelefteq q{\text{ if and only if }}p_{1}+\cdots +p_{k}\leq q_{1}+\cdots +q_{k}{\text{ for all }}k\geq 1.}
1790:
586:
837:
1540:
in the preceding example, their conjugate partitions are and with meet , which is self-conjugate; therefore, the join of
1218:
933:
1106:
36:
32:
1659:
Similarly, there is a dominance order on the set of standard Young bitableaux, which plays a role in the theory of
540:
76:
24:
304:
1785:
84:
1620:
are certain ways to fill Young diagrams with numbers, and a partial order on them (sometimes called the
1564:
560:
88:
1577:
52:
1600:
The dominance order on standard Young tableaux for the partition 6 = 4 + 2
1434:, because the componentwise maximum of two concave sequences need not be concave. For example, for
19:
39:
indicate that the upper node dominates the lower node. While this particular partial ordering is
1712:
1428:
1343:
92:
1672:
1722:
1698:
831:
393:
1584: ≥ 7, it shares some properties with distributive lattices: for example, its
287:
In this definition, partitions are extended by appending zero parts at the end as necessary.
1752:
1736:
1690:
1624:) can be defined in terms of the dominance order on the Young diagrams. For a Young tableau
1551:
667:
371:
654:
637:
633:
1617:
1769:
1757:
1740:
1609:
493:
72:
1677:
812:{\displaystyle {\hat {p}}=(0,p_{1},p_{1}+p_{2},\ldots ,p_{1}+p_{2}+\cdots +p_{n}).}
670:
of this lattice. To explicitly describe the lattice operations, for each partition
379:
40:
1596:
1648:
are first truncated to their sub-tableaux containing entries up to a given value
1563:, such as the minimal height and the maximal covering number, and classified the
43:, this is not true for the dominance ordering on partitions of any number
508:
and then appending it either to the end of the immediately preceding row
1090:{\displaystyle 2{\hat {p}}_{i}\geq {\hat {p}}_{i-1}+{\hat {p}}_{i+1};}
547:′, whose Young diagram is the transpose of the Young diagram of
1716:
1442:
have a join, one uses the conjugation antiautomorphism: the join of
1420:{\displaystyle \operatorname {min} ({\hat {p}}_{i},{\hat {q}}_{i}).}
1323:{\displaystyle {\hat {r}}\leq {\hat {p}},{\hat {r}}\leq {\hat {q}}.}
1595:
18:
1522:{\displaystyle p\lor q=(p^{\prime }\land q^{\prime })^{\prime }.}
374:(and is equivalent to lexicographical ordering) if and only if
147:, with the parts arranged in the weakly decreasing order, then
632:
The dominance ordering determines the inclusions between the
1511:
1501:
1488:
608:
595:
504:
is obtained from it by first removing the last box of row
1640:
as a partition, and moreover the same must hold whenever
922:
are characterized among all integer sequences of length
1179:
By the definition of the dominance ordering, partition
619:{\displaystyle q^{\prime }\trianglelefteq p^{\prime }.}
1195:
is term-by-term less than or equal to the associated (
907:{\displaystyle p_{i}={\hat {p}}_{i}-{\hat {p}}_{i-1}.}
1467:
1360:
1259:
1246:{\displaystyle r\trianglelefteq p,r\trianglelefteq q}
1221:
1109:
1005:
936:
840:
691:
589:
563:
184:
990:{\displaystyle {\hat {p}}_{i}\leq {\hat {p}}_{i+1};}
300:, (1,...,1) is the smallest and (n) is the largest.
1521:
1427:The natural idea to use a similar formula for the
1419:
1322:
1245:
1169:{\displaystyle {\hat {p}}_{0}=0,{\hat {p}}_{n}=n.}
1168:
1089:
989:
926: + 1 by the following three properties:
906:
811:
618:
575:
276:
551:. This operation reverses the dominance ordering:
1554:has determined many invariants of the lattice
8:
1099:The initial term is 0 and the final term is
97:representation theory of the symmetric group
1721:. Vol. 2. Cambridge University Press.
1450:is the conjugate partition of the meet of
492:(Brylawski, Prop. 2.3). Starting from the
1756:
1697:. Oxford University Press. pp. 5–7.
1510:
1500:
1487:
1466:
1405:
1394:
1393:
1383:
1372:
1371:
1359:
1306:
1305:
1291:
1290:
1276:
1275:
1261:
1260:
1258:
1220:
1151:
1140:
1139:
1123:
1112:
1111:
1108:
1072:
1061:
1060:
1044:
1033:
1032:
1022:
1011:
1010:
1004:
972:
961:
960:
950:
939:
938:
935:
889:
878:
877:
867:
856:
855:
845:
839:
797:
778:
765:
746:
733:
720:
693:
692:
690:
666:, and the operation of conjugation is an
607:
594:
588:
562:
260:
254:
235:
222:
203:
194:
183:
1695:Symmetric functions and Hall polynomials
167:is less than or equal to the sum of the
918:+1)-tuples associated to partitions of
386:≤ 6. See image at right for an example.
826:can be recovered from its associated (
657:under the dominance ordering, denoted
1588:takes on only values 0, 1, −1.
1354: + 1)-tuple has components
7:
291:Properties of the dominance ordering
1741:"The lattice of integer partitions"
1628:to dominate another Young tableau
1608:can be graphically represented by
576:{\displaystyle p\trianglelefteq q}
155:in the dominance order if for any
14:
1622:dominance order on Young tableaux
830:+1)-tuple by applying the step 1
23:Example of dominance ordering of
512:− 1, or to the end of row
83:that plays an important role in
1187:if and only if the associated (
303:The dominance ordering implies
91:, especially in the context of
1507:
1480:
1411:
1399:
1377:
1367:
1311:
1296:
1281:
1266:
1145:
1117:
1066:
1038:
1016:
966:
944:
883:
861:
803:
707:
698:
1:
1758:10.1016/0012-365X(73)90094-0
636:of the conjugacy classes of
366:The poset of partitions of
1807:
196: if and only if
1776:Enumerative combinatorics
1718:Enumerative Combinatorics
1199: + 1)-tuple of
1191: + 1)-tuple of
532:all have the same length.
528:of the Young diagram of
500:, the Young diagram of
323:, then for the smallest
305:lexicographical ordering
296:Among the partitions of
143:,...) are partitions of
1781:Algebraic combinatorics
1693:(1979). "section I.1".
1567:of small length. While
1532:For the two partitions
85:algebraic combinatorics
1636:must dominate that of
1601:
1523:
1421:
1324:
1247:
1170:
1091:
991:
908:
813:
620:
577:
278:
79:of a positive integer
48:
1791:Representation theory
1652:, for each choice of
1599:
1524:
1422:
1325:
1248:
1171:
1092:
992:
909:
814:
680: + 1)-tuple
621:
578:
279:
89:representation theory
35:are partitions of 6,
22:
16:Discrete math concept
1745:Discrete Mathematics
1465:
1358:
1346:is the partition of
1257:
1219:
1215:are partitions then
1107:
1003:
934:
838:
689:
587:
561:
543:(or dual) partition
182:
159:≥ 1, the sum of the
53:discrete mathematics
1713:Stanley, Richard P.
1183:precedes partition
262: for all
93:symmetric functions
1661:standard monomials
1602:
1519:
1417:
1350:whose associated (
1320:
1243:
1166:
1087:
987:
904:
809:
638:nilpotent matrices
616:
573:
274:
65:majorization order
61:dominance ordering
49:
47: > 6.
1737:Brylawski, Thomas
1691:Macdonald, Ian G.
1402:
1380:
1314:
1299:
1284:
1269:
1148:
1120:
1069:
1041:
1019:
969:
947:
886:
864:
701:
645:Lattice structure
263:
197:
171:largest parts of
163:largest parts of
1798:
1762:
1760:
1732:
1708:
1616:boxes. Standard
1552:Thomas Brylawski
1528:
1526:
1525:
1520:
1515:
1514:
1505:
1504:
1492:
1491:
1426:
1424:
1423:
1418:
1410:
1409:
1404:
1403:
1395:
1388:
1387:
1382:
1381:
1373:
1329:
1327:
1326:
1321:
1316:
1315:
1307:
1301:
1300:
1292:
1286:
1285:
1277:
1271:
1270:
1262:
1252:
1250:
1249:
1244:
1175:
1173:
1172:
1167:
1156:
1155:
1150:
1149:
1141:
1128:
1127:
1122:
1121:
1113:
1096:
1094:
1093:
1088:
1083:
1082:
1071:
1070:
1062:
1055:
1054:
1043:
1042:
1034:
1027:
1026:
1021:
1020:
1012:
996:
994:
993:
988:
983:
982:
971:
970:
962:
955:
954:
949:
948:
940:
913:
911:
910:
905:
900:
899:
888:
887:
879:
872:
871:
866:
865:
857:
850:
849:
818:
816:
815:
810:
802:
801:
783:
782:
770:
769:
751:
750:
738:
737:
725:
724:
703:
702:
694:
668:antiautomorphism
634:Zariski closures
625:
623:
622:
617:
612:
611:
599:
598:
582:
580:
579:
574:
535:Every partition
372:linearly ordered
283:
281:
280:
275:
264:
261:
259:
258:
240:
239:
227:
226:
208:
207:
198:
195:
69:natural ordering
31: = 6,
1806:
1805:
1801:
1800:
1799:
1797:
1796:
1795:
1766:
1765:
1735:
1729:
1711:
1705:
1689:
1686:
1673:Young's lattice
1669:
1632:, the shape of
1594:
1592:Generalizations
1586:Möbius function
1575:
1562:
1506:
1496:
1483:
1463:
1462:
1392:
1370:
1356:
1355:
1255:
1254:
1253:if and only if
1217:
1216:
1138:
1110:
1105:
1104:
1059:
1031:
1009:
1001:
1000:
959:
937:
932:
931:
930:Nondecreasing,
914:Moreover, the (
876:
854:
841:
836:
835:
793:
774:
761:
742:
729:
716:
687:
686:
665:
647:
603:
590:
585:
584:
583:if and only if
559:
558:
491:
482:
466:and either (1)
453:
444:
435:
426:
417:
408:
400:if and only if
382:if and only if
362:
353:
344:
335:
293:
250:
231:
218:
199:
180:
179:
142:
135:
124:
117:
105:
57:dominance order
17:
12:
11:
5:
1804:
1802:
1794:
1793:
1788:
1786:Lattice theory
1783:
1778:
1768:
1767:
1764:
1763:
1733:
1727:
1709:
1703:
1685:
1682:
1681:
1680:
1675:
1668:
1665:
1618:Young tableaux
1610:Young diagrams
1604:Partitions of
1593:
1590:
1571:
1558:
1530:
1529:
1518:
1513:
1509:
1503:
1499:
1495:
1490:
1486:
1482:
1479:
1476:
1473:
1470:
1416:
1413:
1408:
1401:
1398:
1391:
1386:
1379:
1376:
1369:
1366:
1363:
1319:
1313:
1310:
1304:
1298:
1295:
1289:
1283:
1280:
1274:
1268:
1265:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1177:
1176:
1165:
1162:
1159:
1154:
1147:
1144:
1137:
1134:
1131:
1126:
1119:
1116:
1097:
1086:
1081:
1078:
1075:
1068:
1065:
1058:
1053:
1050:
1047:
1040:
1037:
1030:
1025:
1018:
1015:
1008:
997:
986:
981:
978:
975:
968:
965:
958:
953:
946:
943:
903:
898:
895:
892:
885:
882:
875:
870:
863:
860:
853:
848:
844:
822:The partition
820:
819:
808:
805:
800:
796:
792:
789:
786:
781:
777:
773:
768:
764:
760:
757:
754:
749:
745:
741:
736:
732:
728:
723:
719:
715:
712:
709:
706:
700:
697:
661:
649:Partitions of
646:
643:
642:
641:
629:
628:
627:
626:
615:
610:
606:
602:
597:
593:
572:
569:
566:
553:
552:
533:
487:
478:
449:
440:
431:
422:
413:
404:
387:
364:
358:
349:
340:
331:
301:
292:
289:
285:
284:
273:
270:
267:
257:
253:
249:
246:
243:
238:
234:
230:
225:
221:
217:
214:
211:
206:
202:
193:
190:
187:
140:
133:
122:
115:
104:
101:
75:on the set of
15:
13:
10:
9:
6:
4:
3:
2:
1803:
1792:
1789:
1787:
1784:
1782:
1779:
1777:
1774:
1773:
1771:
1759:
1754:
1750:
1746:
1742:
1738:
1734:
1730:
1728:0-521-56069-1
1724:
1720:
1719:
1714:
1710:
1706:
1704:0-19-853530-9
1700:
1696:
1692:
1688:
1687:
1683:
1679:
1676:
1674:
1671:
1670:
1666:
1664:
1662:
1657:
1655:
1651:
1647:
1643:
1639:
1635:
1631:
1627:
1623:
1619:
1615:
1611:
1607:
1598:
1591:
1589:
1587:
1583:
1579:
1574:
1570:
1566:
1561:
1557:
1553:
1549:
1547:
1543:
1539:
1535:
1516:
1497:
1493:
1484:
1477:
1474:
1471:
1468:
1461:
1460:
1459:
1457:
1453:
1449:
1445:
1441:
1437:
1433:
1430:
1414:
1406:
1396:
1389:
1384:
1374:
1364:
1361:
1353:
1349:
1345:
1341:
1337:
1333:
1317:
1308:
1302:
1293:
1287:
1278:
1272:
1263:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1214:
1210:
1206:
1202:
1198:
1194:
1190:
1186:
1182:
1163:
1160:
1157:
1152:
1142:
1135:
1132:
1129:
1124:
1114:
1102:
1098:
1084:
1079:
1076:
1073:
1063:
1056:
1051:
1048:
1045:
1035:
1028:
1023:
1013:
1006:
998:
984:
979:
976:
973:
963:
956:
951:
941:
929:
928:
927:
925:
921:
917:
901:
896:
893:
890:
880:
873:
868:
858:
851:
846:
842:
833:
829:
825:
806:
798:
794:
790:
787:
784:
779:
775:
771:
766:
762:
758:
755:
752:
747:
743:
739:
734:
730:
726:
721:
717:
713:
710:
704:
695:
685:
684:
683:
681:
679:
674:consider the
673:
669:
664:
660:
656:
652:
644:
639:
635:
631:
630:
613:
604:
600:
591:
570:
567:
564:
557:
556:
555:
554:
550:
546:
542:
538:
534:
531:
527:
523:
519:
515:
511:
507:
503:
499:
495:
494:Young diagram
490:
486:
481:
477:
473:
469:
465:
461:
457:
452:
448:
443:
439:
434:
430:
425:
421:
416:
412:
407:
403:
399:
395:
392:
388:
385:
381:
377:
373:
369:
365:
361:
357:
352:
348:
343:
339:
334:
330:
326:
322:
319: ≠
318:
314:
310:
306:
302:
299:
295:
294:
290:
288:
271:
268:
265:
255:
251:
247:
244:
241:
236:
232:
228:
223:
219:
215:
212:
209:
204:
200:
191:
188:
185:
178:
177:
176:
174:
170:
166:
162:
158:
154:
150:
146:
139:
132:
128:
121:
114:
110:
102:
100:
98:
94:
90:
86:
82:
78:
74:
73:partial order
70:
66:
62:
58:
54:
46:
42:
38:
34:
30:
26:
21:
1751:(3): 201–2.
1748:
1744:
1717:
1694:
1678:Majorization
1660:
1658:
1653:
1649:
1645:
1641:
1637:
1633:
1629:
1625:
1621:
1613:
1605:
1603:
1581:
1578:distributive
1572:
1568:
1559:
1555:
1550:
1545:
1541:
1537:
1533:
1531:
1455:
1454:′ and
1451:
1447:
1443:
1439:
1435:
1431:
1351:
1347:
1339:
1335:
1331:
1212:
1208:
1204:
1200:
1196:
1192:
1188:
1184:
1180:
1178:
1100:
923:
919:
915:
827:
823:
821:
677:
676:associated (
675:
671:
662:
658:
650:
648:
548:
544:
536:
529:
525:
521:
520:if the rows
517:
513:
509:
505:
501:
497:
488:
484:
479:
475:
471:
467:
463:
459:
455:
450:
446:
441:
437:
432:
428:
423:
419:
414:
410:
405:
401:
397:
396:a partition
390:
389:A partition
383:
375:
367:
359:
355:
350:
346:
341:
337:
332:
328:
324:
320:
316:
312:
308:
297:
286:
172:
168:
164:
160:
156:
152:
148:
144:
137:
130:
126:
119:
112:
108:
106:
80:
68:
64:
60:
56:
50:
44:
28:
27:of n. Here,
474:+ 1 or (2)
436:− 1,
378:≤ 5. It is
59:(synonyms:
1770:Categories
1684:References
832:difference
327:such that
311:dominates
307:, i.e. if
125:,...) and
103:Definition
77:partitions
25:partitions
1565:intervals
1512:′
1502:′
1494:∧
1489:′
1472:∨
1458:′:
1400:^
1378:^
1365:
1312:^
1303:≤
1297:^
1282:^
1273:≤
1267:^
1238:⊴
1226:⊴
1146:^
1118:^
1067:^
1049:−
1039:^
1029:≥
1017:^
999:Concave,
967:^
957:≤
945:^
894:−
884:^
874:−
862:^
788:⋯
756:…
699:^
609:′
601:⊴
596:′
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269:≥
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229:≤
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189:⊴
151:precedes
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1715:(1999).
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