Identifier
Identifier
Values
[2] generating graphics... => 2
[1,1] generating graphics... => 1
[3] generating graphics... => 3
[2,1] generating graphics... => 3
[1,1,1] generating graphics... => 1
[4] generating graphics... => 5
[3,1] generating graphics... => 7
[2,2] generating graphics... => 4
[2,1,1] generating graphics... => 4
[1,1,1,1] generating graphics... => 1
[5] generating graphics... => 7
[4,1] generating graphics... => 13
[3,2] generating graphics... => 12
[3,1,1] generating graphics... => 11
[2,2,1] generating graphics... => 8
[2,1,1,1] generating graphics... => 5
[1,1,1,1,1] generating graphics... => 1
[6] generating graphics... => 11
[5,1] generating graphics... => 24
[4,2] generating graphics... => 30
[4,1,1] generating graphics... => 25
[3,3] generating graphics... => 14
[3,2,1] generating graphics... => 33
[3,1,1,1] generating graphics... => 16
[2,2,2] generating graphics... => 9
[2,2,1,1] generating graphics... => 13
[2,1,1,1,1] generating graphics... => 6
[1,1,1,1,1,1] generating graphics... => 1
[7] generating graphics... => 15
[6,1] generating graphics... => 39
[5,2] generating graphics... => 59
[5,1,1] generating graphics... => 50
[4,3] generating graphics... => 47
[4,2,1] generating graphics... => 90
[4,1,1,1] generating graphics... => 41
[3,3,1] generating graphics... => 48
[3,2,2] generating graphics... => 43
[3,2,1,1] generating graphics... => 62
[3,1,1,1,1] generating graphics... => 22
[2,2,2,1] generating graphics... => 22
[2,2,1,1,1] generating graphics... => 19
[2,1,1,1,1,1] generating graphics... => 7
[1,1,1,1,1,1,1] generating graphics... => 1
[8] generating graphics... => 22
[7,1] generating graphics... => 64
[6,2] generating graphics... => 113
[6,1,1] generating graphics... => 94
[5,3] generating graphics... => 119
[5,2,1] generating graphics... => 211
[5,1,1,1] generating graphics... => 92
[4,4] generating graphics... => 53
[4,3,1] generating graphics... => 195
[4,2,2] generating graphics... => 141
[4,2,1,1] generating graphics... => 196
[4,1,1,1,1] generating graphics... => 63
[3,3,2] generating graphics... => 95
[3,3,1,1] generating graphics... => 112
[3,2,2,1] generating graphics... => 130
[3,2,1,1,1] generating graphics... => 103
[3,1,1,1,1,1] generating graphics... => 29
[2,2,2,2] generating graphics... => 23
[2,2,2,1,1] generating graphics... => 41
[2,2,1,1,1,1] generating graphics... => 26
[2,1,1,1,1,1,1] generating graphics... => 8
[1,1,1,1,1,1,1,1] generating graphics... => 1
[9] generating graphics... => 30
[8,1] generating graphics... => 98
[7,2] generating graphics... => 195
[7,1,1] generating graphics... => 164
[6,3] generating graphics... => 250
[6,2,1] generating graphics... => 432
[6,1,1,1] generating graphics... => 187
[5,4] generating graphics... => 183
[5,3,1] generating graphics... => 540
[5,2,2] generating graphics... => 361
[5,2,1,1] generating graphics... => 502
[5,1,1,1,1] generating graphics... => 155
[4,4,1] generating graphics... => 254
[4,3,2] generating graphics... => 440
[4,3,1,1] generating graphics... => 506
[4,2,2,1] generating graphics... => 470
[4,2,1,1,1] generating graphics... => 362
[4,1,1,1,1,1] generating graphics... => 92
[3,3,3] generating graphics... => 97
[3,3,2,1] generating graphics... => 339
[3,3,1,1,1] generating graphics... => 215
[3,2,2,2] generating graphics... => 154
[3,2,2,1,1] generating graphics... => 274
[3,2,1,1,1,1] generating graphics... => 158
[3,1,1,1,1,1,1] generating graphics... => 37
[2,2,2,2,1] generating graphics... => 64
[2,2,2,1,1,1] generating graphics... => 67
[2,2,1,1,1,1,1] generating graphics... => 34
[2,1,1,1,1,1,1,1] generating graphics... => 9
[1,1,1,1,1,1,1,1,1] generating graphics... => 1
[10] generating graphics... => 42
[9,1] generating graphics... => 150
[8,2] generating graphics... => 331
[8,1,1] generating graphics... => 278
[7,3] generating graphics... => 490
[7,2,1] generating graphics... => 835
[7,1,1,1] generating graphics... => 357
[6,4] generating graphics... => 472
[6,3,1] generating graphics... => 1280
[6,2,2] generating graphics... => 829
[6,2,1,1] generating graphics... => 1142
[6,1,1,1,1] generating graphics... => 343
[5,5] generating graphics... => 198
[5,4,1] generating graphics... => 1020
[5,3,2] generating graphics... => 1390
[5,3,1,1] generating graphics... => 1576
[5,2,2,1] generating graphics... => 1359
[5,2,1,1,1] generating graphics... => 1023
[5,1,1,1,1,1] generating graphics... => 247
[4,4,2] generating graphics... => 719
[4,4,1,1] generating graphics... => 773
[4,3,3] generating graphics... => 552
[4,3,2,1] generating graphics... => 1785
[4,3,1,1,1] generating graphics... => 1088
[4,2,2,2] generating graphics... => 636
[4,2,2,1,1] generating graphics... => 1112
[4,2,1,1,1,1] generating graphics... => 612
[4,1,1,1,1,1,1] generating graphics... => 129
[3,3,3,1] generating graphics... => 442
[3,3,2,2] generating graphics... => 500
[3,3,2,1,1] generating graphics... => 833
[3,3,1,1,1,1] generating graphics... => 373
[3,2,2,2,1] generating graphics... => 496
[3,2,2,1,1,1] generating graphics... => 499
[3,2,1,1,1,1,1] generating graphics... => 229
[3,1,1,1,1,1,1,1] generating graphics... => 46
[2,2,2,2,2] generating graphics... => 65
[2,2,2,2,1,1] generating graphics... => 131
[2,2,2,1,1,1,1] generating graphics... => 101
[2,2,1,1,1,1,1,1] generating graphics... => 43
[2,1,1,1,1,1,1,1,1] generating graphics... => 10
[1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[11] generating graphics... => 56
[10,1] generating graphics... => 219
[9,2] generating graphics... => 528
[9,1,1] generating graphics... => 448
[8,3] generating graphics... => 880
[8,2,1] generating graphics... => 1498
[8,1,1,1] generating graphics... => 642
[7,4] generating graphics... => 1018
[7,3,1] generating graphics... => 2682
[7,2,2] generating graphics... => 1707
[7,2,1,1] generating graphics... => 2358
[7,1,1,1,1] generating graphics... => 701
[6,5] generating graphics... => 704
[6,4,1] generating graphics... => 2841
[6,3,2] generating graphics... => 3568
[6,3,1,1] generating graphics... => 4034
[6,2,2,1] generating graphics... => 3359
[6,2,1,1,1] generating graphics... => 2512
[6,1,1,1,1,1] generating graphics... => 590
[5,5,1] generating graphics... => 1246
[5,4,2] generating graphics... => 3182
[5,4,1,1] generating graphics... => 3396
[5,3,3] generating graphics... => 1970
[5,3,2,1] generating graphics... => 6154
[5,3,1,1,1] generating graphics... => 3693
[5,2,2,2] generating graphics... => 2008
[5,2,2,1,1] generating graphics... => 3500
[5,2,1,1,1,1] generating graphics... => 1882
[5,1,1,1,1,1,1] generating graphics... => 376
[4,4,3] generating graphics... => 1287
[4,4,2,1] generating graphics... => 3298
[4,4,1,1,1] generating graphics... => 1864
[4,3,3,1] generating graphics... => 2795
[4,3,2,2] generating graphics... => 2936
[4,3,2,1,1] generating graphics... => 4826
[4,3,1,1,1,1] generating graphics... => 2073
[4,2,2,2,1] generating graphics... => 2248
[4,2,2,1,1,1] generating graphics... => 2223
[4,2,1,1,1,1,1] generating graphics... => 970
[4,1,1,1,1,1,1,1] generating graphics... => 175
[3,3,3,2] generating graphics... => 946
[3,3,3,1,1] generating graphics... => 1277
[3,3,2,2,1] generating graphics... => 1832
[3,3,2,1,1,1] generating graphics... => 1705
[3,3,1,1,1,1,1] generating graphics... => 602
[3,2,2,2,2] generating graphics... => 562
[3,2,2,2,1,1] generating graphics... => 1126
[3,2,2,1,1,1,1] generating graphics... => 829
[3,2,1,1,1,1,1,1] generating graphics... => 318
[3,1,1,1,1,1,1,1,1] generating graphics... => 56
[2,2,2,2,2,1] generating graphics... => 196
[2,2,2,2,1,1,1] generating graphics... => 232
[2,2,2,1,1,1,1,1] generating graphics... => 144
[2,2,1,1,1,1,1,1,1] generating graphics... => 53
[2,1,1,1,1,1,1,1,1,1] generating graphics... => 11
[1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[12] generating graphics... => 77
[11,1] generating graphics... => 322
[10,2] generating graphics... => 838
[10,1,1] generating graphics... => 711
[9,3] generating graphics... => 1539
[9,2,1] generating graphics... => 2608
[9,1,1,1] generating graphics... => 1113
[8,4] generating graphics... => 2046
[8,3,1] generating graphics... => 5285
[8,2,2] generating graphics... => 3332
[8,2,1,1] generating graphics... => 4588
[8,1,1,1,1] generating graphics... => 1350
[7,5] generating graphics... => 1840
[7,4,1] generating graphics... => 6795
[7,3,2] generating graphics... => 8210
[7,3,1,1] generating graphics... => 9237
[7,2,2,1] generating graphics... => 7554
[7,2,1,1,1] generating graphics... => 5603
[7,1,1,1,1,1] generating graphics... => 1292
[6,6] generating graphics... => 751
[6,5,1] generating graphics... => 4962
[6,4,2] generating graphics... => 9862
[6,4,1,1] generating graphics... => 10440
[6,3,3] generating graphics... => 5673
[6,3,2,1] generating graphics... => 17371
[6,3,1,1,1] generating graphics... => 10298
[6,2,2,2] generating graphics... => 5442
[6,2,2,1,1] generating graphics... => 9428
[6,2,1,1,1,1] generating graphics... => 4989
[6,1,1,1,1,1,1] generating graphics... => 966
[5,5,2] generating graphics... => 4543
[5,5,1,1] generating graphics... => 4716
[5,4,3] generating graphics... => 6582
[5,4,2,1] generating graphics... => 16251
[5,4,1,1,1] generating graphics... => 9003
[5,3,3,1] generating graphics... => 11053
[5,3,2,2] generating graphics... => 11225
[5,3,2,1,1] generating graphics... => 18277
[5,3,1,1,1,1] generating graphics... => 7657
[5,2,2,2,1] generating graphics... => 7804
[5,2,2,1,1,1] generating graphics... => 7615
[5,2,1,1,1,1,1] generating graphics... => 3228
[5,1,1,1,1,1,1,1] generating graphics... => 551
[4,4,4] generating graphics... => 1314
[4,4,3,1] generating graphics... => 7466
[4,4,2,2] generating graphics... => 6303
[4,4,2,1,1] generating graphics... => 10042
[4,4,1,1,1,1] generating graphics... => 3942
[4,3,3,2] generating graphics... => 6738
[4,3,3,1,1] generating graphics... => 8944
[4,3,2,2,1] generating graphics... => 11906
[4,3,2,1,1,1] generating graphics... => 10843
[4,3,1,1,1,1,1] generating graphics... => 3645
[4,2,2,2,2] generating graphics... => 2827
[4,2,2,2,1,1] generating graphics... => 5607
[4,2,2,1,1,1,1] generating graphics... => 4022
[4,2,1,1,1,1,1,1] generating graphics... => 1463
[4,1,1,1,1,1,1,1,1] generating graphics... => 231
[3,3,3,3] generating graphics... => 953
[3,3,3,2,1] generating graphics... => 4073
[3,3,3,1,1,1] generating graphics... => 2987
[3,3,2,2,2] generating graphics... => 2405
[3,3,2,2,1,1] generating graphics... => 4672
[3,3,2,1,1,1,1] generating graphics... => 3136
[3,3,1,1,1,1,1,1] generating graphics... => 920
[3,2,2,2,2,1] generating graphics... => 1889
[3,2,2,2,1,1,1] generating graphics... => 2187
[3,2,2,1,1,1,1,1] generating graphics... => 1291
[3,2,1,1,1,1,1,1,1] generating graphics... => 427
[3,1,1,1,1,1,1,1,1,1] generating graphics... => 67
[2,2,2,2,2,2] generating graphics... => 197
[2,2,2,2,2,1,1] generating graphics... => 428
[2,2,2,2,1,1,1,1] generating graphics... => 376
[2,2,2,1,1,1,1,1,1] generating graphics... => 197
[2,2,1,1,1,1,1,1,1,1] generating graphics... => 64
[2,1,1,1,1,1,1,1,1,1,1] generating graphics... => 12
[1,1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
click to show generating function       
Description
The number of semistandard Young tableaux of partition weight of given shape.
The weight of a semistandard Young tableaux is the sequence $(m_1, m_2,\dots)$, where $m_i$ is the number of occurrences of the number $i$ in the tableau. This statistic counts those tableaux whose weight is a weakly decreasing sequence.
Alternatively, this is the sum of the entries in the column specified by the partition of the change of basis matrix from Schur functions to monomial symmetric functions.
References
[1] a(n)= sum of entries of n-th Kostka matrix for the partitions of n. OEIS:A178718
Code
def statistic(mu):
    m = SymmetricFunctions(ZZ).m()
    s = SymmetricFunctions(ZZ).s()
    return sum(coeff for _, coeff in m(s(mu)))

Created
May 20, 2017 at 17:26 by Martin Rubey
Updated
May 20, 2017 at 22:43 by Martin Rubey