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Statistic identifier: St000815
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
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.
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References: [1] a(n)= sum of entries of n-th Kostka matrix for the partitions of n. [[OEIS:A178718]]
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Code:
def statistic(mu):
m = SymmetricFunctions(ZZ).m()
s = SymmetricFunctions(ZZ).s()
return sum(coeff for _, coeff in m(s(mu)))
-----------------------------------------------------------------------------
Statistic values:
[2] => 2
[1,1] => 1
[3] => 3
[2,1] => 3
[1,1,1] => 1
[4] => 5
[3,1] => 7
[2,2] => 4
[2,1,1] => 4
[1,1,1,1] => 1
[5] => 7
[4,1] => 13
[3,2] => 12
[3,1,1] => 11
[2,2,1] => 8
[2,1,1,1] => 5
[1,1,1,1,1] => 1
[6] => 11
[5,1] => 24
[4,2] => 30
[4,1,1] => 25
[3,3] => 14
[3,2,1] => 33
[3,1,1,1] => 16
[2,2,2] => 9
[2,2,1,1] => 13
[2,1,1,1,1] => 6
[1,1,1,1,1,1] => 1
[7] => 15
[6,1] => 39
[5,2] => 59
[5,1,1] => 50
[4,3] => 47
[4,2,1] => 90
[4,1,1,1] => 41
[3,3,1] => 48
[3,2,2] => 43
[3,2,1,1] => 62
[3,1,1,1,1] => 22
[2,2,2,1] => 22
[2,2,1,1,1] => 19
[2,1,1,1,1,1] => 7
[1,1,1,1,1,1,1] => 1
[8] => 22
[7,1] => 64
[6,2] => 113
[6,1,1] => 94
[5,3] => 119
[5,2,1] => 211
[5,1,1,1] => 92
[4,4] => 53
[4,3,1] => 195
[4,2,2] => 141
[4,2,1,1] => 196
[4,1,1,1,1] => 63
[3,3,2] => 95
[3,3,1,1] => 112
[3,2,2,1] => 130
[3,2,1,1,1] => 103
[3,1,1,1,1,1] => 29
[2,2,2,2] => 23
[2,2,2,1,1] => 41
[2,2,1,1,1,1] => 26
[2,1,1,1,1,1,1] => 8
[1,1,1,1,1,1,1,1] => 1
[9] => 30
[8,1] => 98
[7,2] => 195
[7,1,1] => 164
[6,3] => 250
[6,2,1] => 432
[6,1,1,1] => 187
[5,4] => 183
[5,3,1] => 540
[5,2,2] => 361
[5,2,1,1] => 502
[5,1,1,1,1] => 155
[4,4,1] => 254
[4,3,2] => 440
[4,3,1,1] => 506
[4,2,2,1] => 470
[4,2,1,1,1] => 362
[4,1,1,1,1,1] => 92
[3,3,3] => 97
[3,3,2,1] => 339
[3,3,1,1,1] => 215
[3,2,2,2] => 154
[3,2,2,1,1] => 274
[3,2,1,1,1,1] => 158
[3,1,1,1,1,1,1] => 37
[2,2,2,2,1] => 64
[2,2,2,1,1,1] => 67
[2,2,1,1,1,1,1] => 34
[2,1,1,1,1,1,1,1] => 9
[1,1,1,1,1,1,1,1,1] => 1
[10] => 42
[9,1] => 150
[8,2] => 331
[8,1,1] => 278
[7,3] => 490
[7,2,1] => 835
[7,1,1,1] => 357
[6,4] => 472
[6,3,1] => 1280
[6,2,2] => 829
[6,2,1,1] => 1142
[6,1,1,1,1] => 343
[5,5] => 198
[5,4,1] => 1020
[5,3,2] => 1390
[5,3,1,1] => 1576
[5,2,2,1] => 1359
[5,2,1,1,1] => 1023
[5,1,1,1,1,1] => 247
[4,4,2] => 719
[4,4,1,1] => 773
[4,3,3] => 552
[4,3,2,1] => 1785
[4,3,1,1,1] => 1088
[4,2,2,2] => 636
[4,2,2,1,1] => 1112
[4,2,1,1,1,1] => 612
[4,1,1,1,1,1,1] => 129
[3,3,3,1] => 442
[3,3,2,2] => 500
[3,3,2,1,1] => 833
[3,3,1,1,1,1] => 373
[3,2,2,2,1] => 496
[3,2,2,1,1,1] => 499
[3,2,1,1,1,1,1] => 229
[3,1,1,1,1,1,1,1] => 46
[2,2,2,2,2] => 65
[2,2,2,2,1,1] => 131
[2,2,2,1,1,1,1] => 101
[2,2,1,1,1,1,1,1] => 43
[2,1,1,1,1,1,1,1,1] => 10
[1,1,1,1,1,1,1,1,1,1] => 1
[11] => 56
[10,1] => 219
[9,2] => 528
[9,1,1] => 448
[8,3] => 880
[8,2,1] => 1498
[8,1,1,1] => 642
[7,4] => 1018
[7,3,1] => 2682
[7,2,2] => 1707
[7,2,1,1] => 2358
[7,1,1,1,1] => 701
[6,5] => 704
[6,4,1] => 2841
[6,3,2] => 3568
[6,3,1,1] => 4034
[6,2,2,1] => 3359
[6,2,1,1,1] => 2512
[6,1,1,1,1,1] => 590
[5,5,1] => 1246
[5,4,2] => 3182
[5,4,1,1] => 3396
[5,3,3] => 1970
[5,3,2,1] => 6154
[5,3,1,1,1] => 3693
[5,2,2,2] => 2008
[5,2,2,1,1] => 3500
[5,2,1,1,1,1] => 1882
[5,1,1,1,1,1,1] => 376
[4,4,3] => 1287
[4,4,2,1] => 3298
[4,4,1,1,1] => 1864
[4,3,3,1] => 2795
[4,3,2,2] => 2936
[4,3,2,1,1] => 4826
[4,3,1,1,1,1] => 2073
[4,2,2,2,1] => 2248
[4,2,2,1,1,1] => 2223
[4,2,1,1,1,1,1] => 970
[4,1,1,1,1,1,1,1] => 175
[3,3,3,2] => 946
[3,3,3,1,1] => 1277
[3,3,2,2,1] => 1832
[3,3,2,1,1,1] => 1705
[3,3,1,1,1,1,1] => 602
[3,2,2,2,2] => 562
[3,2,2,2,1,1] => 1126
[3,2,2,1,1,1,1] => 829
[3,2,1,1,1,1,1,1] => 318
[3,1,1,1,1,1,1,1,1] => 56
[2,2,2,2,2,1] => 196
[2,2,2,2,1,1,1] => 232
[2,2,2,1,1,1,1,1] => 144
[2,2,1,1,1,1,1,1,1] => 53
[2,1,1,1,1,1,1,1,1,1] => 11
[1,1,1,1,1,1,1,1,1,1,1] => 1
[12] => 77
[11,1] => 322
[10,2] => 838
[10,1,1] => 711
[9,3] => 1539
[9,2,1] => 2608
[9,1,1,1] => 1113
[8,4] => 2046
[8,3,1] => 5285
[8,2,2] => 3332
[8,2,1,1] => 4588
[8,1,1,1,1] => 1350
[7,5] => 1840
[7,4,1] => 6795
[7,3,2] => 8210
[7,3,1,1] => 9237
[7,2,2,1] => 7554
[7,2,1,1,1] => 5603
[7,1,1,1,1,1] => 1292
[6,6] => 751
[6,5,1] => 4962
[6,4,2] => 9862
[6,4,1,1] => 10440
[6,3,3] => 5673
[6,3,2,1] => 17371
[6,3,1,1,1] => 10298
[6,2,2,2] => 5442
[6,2,2,1,1] => 9428
[6,2,1,1,1,1] => 4989
[6,1,1,1,1,1,1] => 966
[5,5,2] => 4543
[5,5,1,1] => 4716
[5,4,3] => 6582
[5,4,2,1] => 16251
[5,4,1,1,1] => 9003
[5,3,3,1] => 11053
[5,3,2,2] => 11225
[5,3,2,1,1] => 18277
[5,3,1,1,1,1] => 7657
[5,2,2,2,1] => 7804
[5,2,2,1,1,1] => 7615
[5,2,1,1,1,1,1] => 3228
[5,1,1,1,1,1,1,1] => 551
[4,4,4] => 1314
[4,4,3,1] => 7466
[4,4,2,2] => 6303
[4,4,2,1,1] => 10042
[4,4,1,1,1,1] => 3942
[4,3,3,2] => 6738
[4,3,3,1,1] => 8944
[4,3,2,2,1] => 11906
[4,3,2,1,1,1] => 10843
[4,3,1,1,1,1,1] => 3645
[4,2,2,2,2] => 2827
[4,2,2,2,1,1] => 5607
[4,2,2,1,1,1,1] => 4022
[4,2,1,1,1,1,1,1] => 1463
[4,1,1,1,1,1,1,1,1] => 231
[3,3,3,3] => 953
[3,3,3,2,1] => 4073
[3,3,3,1,1,1] => 2987
[3,3,2,2,2] => 2405
[3,3,2,2,1,1] => 4672
[3,3,2,1,1,1,1] => 3136
[3,3,1,1,1,1,1,1] => 920
[3,2,2,2,2,1] => 1889
[3,2,2,2,1,1,1] => 2187
[3,2,2,1,1,1,1,1] => 1291
[3,2,1,1,1,1,1,1,1] => 427
[3,1,1,1,1,1,1,1,1,1] => 67
[2,2,2,2,2,2] => 197
[2,2,2,2,2,1,1] => 428
[2,2,2,2,1,1,1,1] => 376
[2,2,2,1,1,1,1,1,1] => 197
[2,2,1,1,1,1,1,1,1,1] => 64
[2,1,1,1,1,1,1,1,1,1,1] => 12
[1,1,1,1,1,1,1,1,1,1,1,1] => 1
-----------------------------------------------------------------------------
Created: May 20, 2017 at 17:26 by Martin Rubey
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Last Updated: May 20, 2017 at 22:43 by Martin Rubey