1138:
177:
375:
169:
461:
590:
754:
While the map from partitions to their Young diagrams is injective, this is not the case for the map from skew shapes to skew diagrams; therefore the shape of a skew diagram cannot always be determined from the set of filled squares only. Although many properties of skew tableaux only depend on the filled squares, some operations defined on them do require explicit knowledge of
1801:
272:; however, French notation differs from that convention by placing the vertical coordinate first. The figure on the right shows, using the English notation, the Young diagram corresponding to the partition (5, 4, 1) of the number 10. The conjugate partition, measuring the column lengths, is (3, 2, 2, 2, 1).
267:
advises readers preferring the French convention to "read this book upside down in a mirror" (Macdonald 1979, p. 2). This nomenclature probably started out as jocular. The
English notation corresponds to the one universally used for matrices, while the French notation is closer to the convention
753:
is obtained by filling the squares of the corresponding skew diagram; such a tableau is semistandard if entries increase weakly along each row, and increase strictly down each column, and it is standard if moreover all numbers from 1 to the number of squares of the skew diagram occur exactly once.
242:
There is almost universal agreement that in labeling boxes of Young diagrams by pairs of integers, the first index selects the row of the diagram, and the second index selects the box within the row. Nevertheless, two distinct conventions exist to display these diagrams, and consequently tableaux:
1049:
are built. Many facts about a representation can be deduced from the corresponding diagram. Below, we describe two examples: determining the dimension of a representation and restricted representations. In both cases, we will see that some properties of a representation can be determined by using
766:, so it is important that skew tableaux do record this information: two distinct skew tableaux may differ only in their shape, while they occupy the same set of squares, each filled with the same entries. Young tableaux can be identified with skew tableaux in which
1446:
1640:
1614:
2079:, but also in (infinitely) many other ways. In general any skew diagram whose set of non-empty rows (or of non-empty columns) is not contiguous or does not contain the first row (respectively column) will be associated to more than one skew shape.
1327:
1225:
is the number of boxes that are in the same row to the right of it plus those boxes in the same column below it, plus one (for the box itself). By the hook-length formula, the dimension of an irreducible representation is
192:, particularly when represented using dots) is a finite collection of boxes, or cells, arranged in left-justified rows, with the row lengths in non-increasing order. Listing the number of boxes in each row gives a
1338:
573:
There are several variations of this definition: for example, in a row-strict tableau the entries strictly increase along the rows and weakly increase down the columns. Also, tableaux with
1924:
by removing just one box (which must be at the end both of its row and of its column); the restricted representation then decomposes as a direct sum of the irreducible representations of
837:, and it is in fact possible to define (skew) semistandard tableaux as such sequences, as is done by Macdonald (Macdonald 1979, p. 4). This definition incorporates the partitions
1796:{\displaystyle \dim W(\lambda )={\frac {7\cdot 8\cdot 9\cdot 10\cdot 11\cdot 6\cdot 7\cdot 8\cdot 9\cdot 5}{7\cdot 5\cdot 4\cdot 3\cdot 1\cdot 5\cdot 3\cdot 2\cdot 1\cdot 1}}=66528.}
478:, if the entries weakly increase along each row and strictly increase down each column. Recording the number of times each number appears in a tableau gives a sequence known as the
1137:
1497:
1239:
539:
559:
509:
1184:
is equal to the number of different standard Young tableaux that can be obtained from the diagram of the representation. This number can be calculated by the
581:. There are also generalizations such as domino tableaux or ribbon tableaux, in which several boxes may be grouped together before assigning entries to them.
1121:
play a central role, rather than standard tableaux; in particular it is the number of those tableaux that determines the dimension of the representation.
243:
the first places each row below the previous one, the second stacks each row on top of the previous one. Since the former convention is mainly used by
482:
of the tableau. Thus the standard Young tableaux are precisely the semistandard tableaux of weight (1,1,...,1), which requires every integer up to
2124:
have the same (empty) set of entries; for skew tableaux however such distinction is necessary even in cases where the set of entries is not empty.
2505:
881:
827:. Any pair of successive shapes in such a sequence is a skew shape whose diagram contains at most one box in each column; such shapes are called
2138:
1050:
just its diagram. Young tableaux are involved in the use of the symmetric group in quantum chemistry studies of atoms, molecules and solids.
2425:
1046:
1024:
470:
In other applications, it is natural to allow the same number to appear more than once (or not at all) in a tableau. A tableau is called
455:
1627:
the column of a box. For instance, for the partition (5,4,1) we get as dimension of the corresponding irreducible representation of
422:
2241:
1945:
2065:
For instance the skew diagram consisting of a single square at position (2,4) can be obtained by removing the diagram of
1897:
The question of determining this decomposition of the restricted representation of a given irreducible representation of
1441:{\displaystyle \dim \pi _{\lambda }={\frac {10!}{7\cdot 5\cdot 4\cdot 3\cdot 1\cdot 5\cdot 3\cdot 2\cdot 1\cdot 1}}=288.}
2389:
2281:
2206:
2145:
1974:
1918:, is answered as follows. One forms the set of all Young diagrams that can be obtained from the diagram of shape
2213:
2194:
978:
213:, and it carries the same information as that partition. Containment of one Young diagram in another defines a
130:
1332:
The figure on the right shows hook-lengths for all boxes in the diagram of the partition 10 = 5 + 4 + 1. Thus
869:. Various ways of counting Young tableaux have been explored and lead to the definition of and identities for
2409:
2225:
1871:
435:
if the entries in each row and each column are increasing. The number of distinct standard Young tableaux on
142:
2510:
791:
with positive integer entries gives rise to a sequence of partitions (or Young diagrams), by starting with
106:
1609:{\displaystyle \dim W(\lambda )=\prod _{(i,j)\in Y(\lambda )}{\frac {r+j-i}{\operatorname {hook} (i,j)}},}
939:
are parametrized by the set of semistandard Young tableaux of a fixed shape over the alphabet {1, 2, ...,
2515:
2404:
2276:
Oxford
Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 1979. viii+180 pp.
954:
239:; one obtains a Young diagram of that shape by reflecting the original diagram along its main diagonal.
2217:
1030:
922:
1887:
1322:{\displaystyle \dim \pi _{\lambda }={\frac {n!}{\prod _{x\in Y(\lambda )}\operatorname {hook} (x)}}.}
2399:
2317:
918:
114:
1950:
1891:
244:
1233:
divided by the product of the hook lengths of all boxes in the diagram of the representation:
1095:
862:
421:..., but now one usually uses a set of numbers for brevity. In their original application to
269:
225:. Listing the number of boxes of a Young diagram in each column gives another partition, the
86:
2444:
2170:
2344:
2251:
1074:
1054:
926:
102:
94:
2352:
2287:
996:
into irreducible components is formulated in terms of certain skew semistandard tableaux.
888:
product on the set of all semistandard Young tableaux, giving it the structure called the
8:
1185:
1130:
1012:
518:
391:
176:
158:
Note: this article uses the
English convention for displaying Young diagrams and tableaux
966:
866:
544:
494:
260:
222:
146:
2313:"A probabilistic proof of a formula for the number of Young tableaux of a given shape"
386:
is obtained by filling in the boxes of the Young diagram with symbols taken from some
2385:
2335:
2312:
2277:
2255:
2237:
2202:
2141:
2048:
1042:
1000:
442:
431:
distinct entries, arbitrarily assigned to boxes of the diagram. A tableau is called
207:, the total number of boxes of the diagram. The Young diagram is said to be of shape
193:
90:
2417:
2462:
2348:
2330:
2304:
2270:
2229:
2038:
2007:
1008:
946:
870:
374:
264:
214:
30:
1053:
Young diagrams also parametrize the irreducible polynomial representations of the
168:
2469:
2340:
2284:
2247:
1034:
970:
962:
908:
578:
218:
189:
134:
118:
98:
2027:"On the eigenvalues of representations of reflection groups and wreath products"
876:
Many combinatorial algorithms on tableaux are known, including Schützenberger's
2475:
2395:
1038:
1004:
982:
974:
950:
890:
126:
122:
2233:
2011:
803:
places further in the sequence the one whose diagram is obtained from that of
121:
in 1903. Their theory was further developed by many mathematicians, including
2499:
2300:
2259:
2052:
2043:
2026:
877:
858:
281:
138:
110:
82:
1145:
of the boxes for the partition 10 = 5 + 4 + 1
2485:
2308:
1970:
958:
460:
2377:
1936:
corresponding to those diagrams, each occurring exactly once in the sum.
885:
248:
117:, in 1900. They were then applied to the study of the symmetric group by
20:
1115:
nonempty rows). In these cases semistandard tableaux with entries up to
105:
groups and to study their properties. Young tableaux were introduced by
2267:
Howard Georgi, Lie
Algebras in Particle Physics, 2nd Edition - Westview
2199:
Young
Tableaux, with Applications to Representation Theory and Geometry
589:
2228:, Readings in Mathematics. Vol. 129. New York: Springer-Verlag.
2364:
2088:
A somewhat similar situation arises for matrices: the 3-by-0 matrix
1995:
251:, it is customary to refer to these conventions respectively as the
2360:
2326:
1094:
nonempty rows), or the irreducible complex representations of the
1976:
The Art of
Computer Programming, Vol. III: Sorting and Searching
2294:
Symmetric
Functions, Schubert Polynomials, and Degeneracy Loci
1483:
parts) is the number of semistandard Young tableaux of shape
593:
Skew tableau of shape (5, 4, 2, 2) / (2, 1), English notation
448:
1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, ... (sequence
2489:
2479:
1886:. These representations are then called the factors of the
1451:
Similarly, the dimension of the irreducible representation
450:
69:
48:
1073:
nonempty rows), or the irreducible representations of the
901:
In representation theory, standard Young tableaux of size
809:
by adding all the boxes that contain a value ≤
394:. Originally that alphabet was a set of indexed variables
1981:
Such arrangements were introduced by Alfred Young in 1900
715:
is the set-theoretic difference of the Young diagrams of
60:
36:
2174:
1018:
772:
is the empty partition (0) (the unique partition of 0).
577:
entries have been considered, notably, in the theory of
2449:
Group Theory: Birdtracks, Lie's, and
Exceptional Groups
2176:
Group Theory: Birdtracks, Lie's, and
Exceptional Groups
2369:
1643:
1500:
1341:
1242:
1029:
Young diagrams are in one-to-one correspondence with
981:
describing (among other things) the decomposition of
907:
describe bases in irreducible representations of the
547:
521:
497:
306:
in the diagram λ in English notation. Similarly, the
1874:
of several representations that are irreducible for
831:. This sequence of partitions completely determines
66:
63:
57:
45:
42:
33:
2365:
A direct bijective proof of the Hook-length formula
2299:
1828:is also a representation of the symmetric group on
727:: the set of squares that belong to the diagram of
54:
39:
2169:
1795:
1608:
1440:
1321:
1041:. They provide a convenient way of specifying the
821:; this partition eventually becomes equal to
553:
533:
503:
172:Young diagram of shape (5, 4, 1), English notation
999:Applications to algebraic geometry center around
217:on the set of all partitions, which is in fact a
180:Young diagram of shape (5, 4, 1), French notation
2497:
2037:(2). Mathematical Sciences Publishers: 353–396.
1150:The dimension of the irreducible representation
1124:
464:All standard Young tableaux with at most 5 boxes
280:In many applications, for example when defining
93:. It provides a convenient way to describe the
16:A combinatorial object in representation theory
2000:Proceedings of the London Mathematical Society
1491:), which is given by the hook-length formula:
677:, then the containment of diagrams means that
2094:must be distinguished from the 0-by-3 matrix
1806:
857:Young tableaux have numerous applications in
2426:Notices of the American Mathematical Society
2212:
1979:(2nd ed.), Addison-Wesley, p. 48,
1847:. However, an irreducible representation of
1025:Representation theory of the symmetric group
1811:A representation of the symmetric group on
852:
378:A standard Young tableau of shape (5, 4, 1)
344:itself; in other words, the hook length is
2024:
1136:
1015:and described in terms of Young tableaux.
2472:." From MathWorld—A Wolfram Web Resource.
2465:". From MathWorld—A Wolfram Web Resource.
2334:
2274:Symmetric functions and Hall polynomials.
2042:
1996:"On quantitative substitutional analysis"
849:in the data comprising the skew tableau.
491:In a standard Young tableau, the integer
2158:The Symmetric Group in Quantum Chemistry
884:. Lascoux and Schützenberger studied an
588:
561:. The sum of the descents is called the
459:
373:
175:
167:
2394:
2137:, 2nd ed. NRC Research Press, Ottawa
2133:Philip R. Bunker and Per Jensen (1998)
1487:(containing only the entries from 1 to
945:}. This has important consequences for
882:Robinson–Schensted–Knuth correspondence
340:in English notation, including the box
332:is the number of boxes to the right of
302:as the number of boxes to the right of
247:while the latter is often preferred by
2506:Representation theory of finite groups
2498:
423:representations of the symmetric group
2222:Representation theory. A first course
1993:
1969:
1019:Applications in representation theory
275:
2415:
2201:. Cambridge University Press, 1997,
2018:
390:, which is usually required to be a
2451:. Princeton University Press, 2008.
2135:Molecular Symmetry and Spectroscopy
13:
985:of irreducible representations of
14:
2527:
2455:
2416:Yong, Alexander (February 2007).
2296:. American Mathematical Society.
1946:Robinson–Schensted correspondence
613:) such that the Young diagram of
284:, it is convenient to define the
2160:, CRC Press, Boca Raton, Florida
2106:is a 3-by-3 (zero) matrix while
2015:. See in particular p. 133.
1634:(traversing the boxes by rows):
584:
541:appears in a row strictly below
29:
2112:is the 0-by-0 matrix, but both
2025:Stembridge, John (1989-12-01).
1906:, corresponding to a partition
1471:corresponding to the partition
797:, and taking for the partition
320:) is the number of boxes below
259:; for instance, in his book on
2363:, Alexander V. Stoyanovskii, "
2163:
2150:
2127:
2082:
2059:
2031:Pacific Journal of Mathematics
1987:
1963:
1659:
1653:
1597:
1585:
1554:
1548:
1539:
1527:
1516:
1510:
1310:
1304:
1293:
1287:
1108:(again when they have at most
775:Any skew semistandard tableau
619:contains the Young diagram of
152:
1:
2226:Graduate Texts in Mathematics
2188:
2179:. Princeton University Press.
1172:corresponding to a partition
1125:Dimension of a representation
568:
2418:"What is...a Young Tableau?"
2336:10.1016/0001-8708(79)90023-9
949:, starting from the work of
7:
2405:Encyclopedia of Mathematics
2181:, eq. 9.28 and appendix B.4
1939:
1858:may not be irreducible for
1047:irreducible representations
1031:irreducible representations
955:homogeneous coordinate ring
369:
163:
10:
2532:
1807:Restricted representations
1128:
1022:
979:Littlewood–Richardson rule
923:irreducible representation
201:of a non-negative integer
143:Marcel-Paul Schützenberger
2359:Jean-Christophe Novelli,
2234:10.1007/978-1-4612-0979-9
1888:restricted representation
601:is a pair of partitions (
2044:10.2140/pjm.1989.140.353
1956:
1087:(when they have at most
1067:(when they have at most
961:and further explored by
921:in a finite-dimensional
853:Overview of applications
441:entries is given by the
2374:(1997), pp. 53–67.
2318:Advances in Mathematics
2012:10.1112/plms/s1-33.1.97
1870:. Instead, it may be a
1161:of the symmetric group
919:standard monomial basis
488:to occur exactly once.
2070: = (5,3,2,1)
1892:induced representation
1797:
1610:
1442:
1323:
1011:can be represented by
594:
555:
535:
505:
465:
425:, Young tableaux have
379:
181:
173:
2476:Semistandard tableaux
1798:
1611:
1443:
1324:
1096:special unitary group
1003:on Grassmannians and
863:representation theory
592:
556:
536:
506:
463:
377:
270:Cartesian coordinates
179:
171:
95:group representations
87:representation theory
2468:Eric W. Weisstein. "
2461:Eric W. Weisstein. "
1641:
1498:
1339:
1240:
1075:special linear group
1055:general linear group
1013:Schubert polynomials
1007:. Certain important
965:with collaborators,
927:general linear group
545:
519:
495:
221:structure, known as
115:Cambridge University
2511:Symmetric functions
2382:The Symmetric Group
1186:hook length formula
1131:Hook length formula
896:le monoïde plaxique
733:but not to that of
625:; it is denoted by
534:{\displaystyle k+1}
392:totally ordered set
261:symmetric functions
2516:Integer partitions
2445:Predrag Cvitanović
2384:. Springer, 2001,
2171:Predrag Cvitanović
1994:Young, A. (1900),
1951:Schur–Weyl duality
1793:
1623:gives the row and
1606:
1558:
1438:
1319:
1297:
1043:Young symmetrizers
1009:cohomology classes
867:algebraic geometry
595:
551:
531:
501:
466:
443:involution numbers
380:
276:Arm and leg length
182:
174:
147:Richard P. Stanley
2486:Standard tableaux
2305:Nijenhuis, Albert
2292:Laurent Manivel.
2243:978-0-387-97495-8
1785:
1601:
1522:
1430:
1314:
1273:
1208:in Young diagram
1001:Schubert calculus
829:horizontal strips
554:{\displaystyle k}
504:{\displaystyle k}
131:G. de B. Robinson
91:Schubert calculus
85:object useful in
2523:
2441:
2439:
2438:
2422:
2412:
2356:
2338:
2325:(1). Amsterdam:
2309:Wilf, Herbert S.
2271:Macdonald, I. G.
2263:
2182:
2180:
2167:
2161:
2156:R.Pauncz (1995)
2154:
2148:
2131:
2125:
2123:
2117:
2111:
2105:
2099:
2093:
2086:
2080:
2078:
2072:from the one of
2071:
2063:
2057:
2056:
2046:
2022:
2016:
2014:
1991:
1985:
1983:
1971:Knuth, Donald E.
1967:
1935:
1923:
1917:
1911:
1885:
1869:
1857:
1846:
1834:
1827:
1816:
1802:
1800:
1799:
1794:
1786:
1784:
1725:
1666:
1633:
1619:where the index
1615:
1613:
1612:
1607:
1602:
1600:
1577:
1560:
1557:
1470:
1461:
1447:
1445:
1444:
1439:
1431:
1429:
1370:
1362:
1357:
1356:
1328:
1326:
1325:
1320:
1315:
1313:
1296:
1271:
1263:
1258:
1257:
1232:
1224:
1218:
1207:
1201:
1183:
1177:
1171:
1160:
1154:
1140:
1120:
1114:
1107:
1093:
1086:
1072:
1066:
995:
947:invariant theory
944:
938:
916:
906:
848:
842:
836:
826:
820:
814:
808:
802:
796:
790:
780:
771:
765:
759:
752:
738:
732:
726:
720:
714:
705:of a skew shape
700:
696:
676:
655:
634:
624:
618:
612:
606:
579:plane partitions
565:of the tableau.
560:
558:
557:
552:
540:
538:
537:
532:
510:
508:
507:
502:
487:
453:
440:
430:
420:
411:
402:
253:English notation
238:
215:partial ordering
212:
206:
200:
76:
75:
72:
71:
68:
65:
62:
59:
56:
51:
50:
47:
44:
41:
38:
35:
2531:
2530:
2526:
2525:
2524:
2522:
2521:
2520:
2496:
2495:
2463:Ferrers Diagram
2458:
2436:
2434:
2420:
2400:"Young tableau"
2244:
2214:Fulton, William
2191:
2186:
2185:
2168:
2164:
2155:
2151:
2132:
2128:
2119:
2113:
2107:
2101:
2095:
2089:
2087:
2083:
2073:
2066:
2064:
2060:
2023:
2019:
1992:
1988:
1968:
1964:
1959:
1942:
1934:
1925:
1919:
1913:
1907:
1905:
1884:
1875:
1868:
1859:
1856:
1848:
1845:
1836:
1829:
1826:
1818:
1812:
1809:
1726:
1667:
1665:
1642:
1639:
1638:
1632:
1628:
1578:
1561:
1559:
1526:
1499:
1496:
1495:
1469:
1463:
1452:
1371:
1363:
1361:
1352:
1348:
1340:
1337:
1336:
1277:
1272:
1264:
1262:
1253:
1249:
1241:
1238:
1237:
1227:
1220:
1209:
1203:
1195:
1179:
1173:
1170:
1162:
1159:
1152:
1151:
1148:
1147:
1146:
1133:
1127:
1116:
1109:
1106:
1098:
1088:
1085:
1077:
1068:
1065:
1057:
1045:from which the
1039:complex numbers
1035:symmetric group
1027:
1021:
994:
986:
983:tensor products
963:Gian-Carlo Rota
940:
937:
929:
912:
909:symmetric group
902:
871:Schur functions
855:
844:
838:
832:
822:
816:
810:
804:
798:
792:
782:
776:
767:
761:
755:
744:
734:
728:
722:
716:
706:
698:
695:
686:
678:
674:
667:
657:
653:
646:
636:
626:
620:
614:
608:
602:
587:
571:
546:
543:
542:
520:
517:
516:
496:
493:
492:
483:
449:
436:
426:
419:
413:
410:
404:
401:
395:
372:
361:
350:
315:
293:
278:
257:French notation
234:
223:Young's lattice
208:
202:
196:
190:Ferrers diagram
188:(also called a
166:
155:
135:Gian-Carlo Rota
119:Georg Frobenius
53:
32:
28:
17:
12:
11:
5:
2529:
2519:
2518:
2513:
2508:
2494:
2493:
2483:
2473:
2466:
2457:
2456:External links
2454:
2453:
2452:
2442:
2413:
2392:
2378:Bruce E. Sagan
2375:
2357:
2301:Greene, Curtis
2297:
2290:
2268:
2265:
2242:
2210:
2195:William Fulton
2190:
2187:
2184:
2183:
2162:
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2081:
2058:
2017:
1986:
1961:
1960:
1958:
1955:
1954:
1953:
1948:
1941:
1938:
1929:
1901:
1879:
1863:
1852:
1840:
1822:
1808:
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1789:
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1771:
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1753:
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1479:(with at most
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1134:
1129:Main article:
1126:
1123:
1102:
1081:
1061:
1020:
1017:
1005:flag varieties
990:
933:
891:plactic monoid
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586:
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530:
527:
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368:
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313:
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282:Jack functions
277:
274:
165:
162:
154:
151:
127:W. V. D. Hodge
123:Percy MacMahon
103:general linear
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4:
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2488:entry in the
2487:
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2478:entry in the
2477:
2474:
2471:
2470:Young Tableau
2467:
2464:
2460:
2459:
2450:
2446:
2443:
2432:
2428:
2427:
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2396:Vinberg, E.B.
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2390:0-387-95067-2
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2146:9780660196282
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2069:
2062:
2054:
2050:
2045:
2040:
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2009:
2006:(1): 97–145,
2005:
2001:
1997:
1990:
1982:
1978:
1977:
1972:
1966:
1962:
1952:
1949:
1947:
1944:
1943:
1937:
1932:
1928:
1922:
1916:
1910:
1904:
1900:
1895:
1893:
1889:
1882:
1878:
1873:
1866:
1862:
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1843:
1839:
1832:
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1815:
1790:
1787:
1781:
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1757:
1754:
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1727:
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1080:
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1002:
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989:
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924:
920:
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897:
893:
892:
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878:jeu de taquin
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859:combinatorics
850:
847:
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830:
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779:
773:
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690:
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664:
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650:
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639:
633:
629:
623:
617:
611:
605:
600:
591:
585:Skew tableaux
582:
580:
576:
566:
564:
548:
528:
525:
522:
514:
498:
489:
486:
481:
477:
476:column strict
473:
462:
457:
452:
447:
446:
445:
444:
439:
434:
429:
424:
416:
407:
398:
393:
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385:
384:Young tableau
376:
367:
365:
358:
354:
347:
343:
339:
335:
331:
327:
323:
319:
312:
309:
305:
301:
297:
290:
287:
283:
273:
271:
266:
262:
258:
254:
250:
246:
240:
237:
233:partition of
232:
228:
224:
220:
216:
211:
205:
199:
195:
191:
187:
186:Young diagram
178:
170:
161:
159:
150:
148:
144:
140:
139:Alain Lascoux
136:
132:
128:
124:
120:
116:
112:
111:mathematician
108:
104:
100:
96:
92:
88:
84:
83:combinatorial
80:
74:
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25:Young tableau
22:
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2435:. Retrieved
2433:(2): 240–241
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2424:
2403:
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2371:
2368:
2322:
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2020:
2003:
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1989:
1980:
1975:
1965:
1930:
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1920:
1914:
1908:
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741:skew tableau
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703:skew diagram
702:
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157:
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107:Alfred Young
78:
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2329:: 104–109.
2218:Harris, Joe
2140:pp.198-202.
2077:= (5,4,2,1)
1193:hook length
886:associative
563:major index
326:hook length
298:) of a box
245:Anglophones
153:Definitions
21:mathematics
2500:Categories
2437:2008-01-16
2353:0398.05008
2189:References
1890:(see also
1872:direct sum
1835:elements,
1817:elements,
1023:See also:
967:de Concini
599:skew shape
575:decreasing
569:Variations
308:leg length
286:arm length
77:; plural:
2410:EMS Press
2398:(2001) ,
2264:Lecture 4
2260:246650103
2053:0030-8730
1779:⋅
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894:(French:
781:of shape
743:of shape
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231:transpose
227:conjugate
194:partition
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2327:Elsevier
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2173:(2008).
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370:Tableaux
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164:Diagrams
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