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-----------------------------------------------------------------------------
Statistic identifier: St000813
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition.
This is also the sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to monomial symmetric functions.
-----------------------------------------------------------------------------
References: [1] Number of different 0-1 matrices in which the number of 1's is n, with at least one 1 in each row and column. [[OEIS:A068313]]
-----------------------------------------------------------------------------
Code:
def statistic(mu):
m = SymmetricFunctions(ZZ).m()
e = SymmetricFunctions(ZZ).e()
return sum(coeff for _, coeff in m(e(mu)))
-----------------------------------------------------------------------------
Statistic values:
[2] => 1
[1,1] => 3
[3] => 1
[2,1] => 4
[1,1,1] => 10
[4] => 1
[3,1] => 5
[2,2] => 9
[2,1,1] => 20
[1,1,1,1] => 47
[5] => 1
[4,1] => 6
[3,2] => 14
[3,1,1] => 30
[2,2,1] => 50
[2,1,1,1] => 110
[1,1,1,1,1] => 246
[6] => 1
[5,1] => 7
[4,2] => 20
[4,1,1] => 42
[3,3] => 29
[3,2,1] => 97
[3,1,1,1] => 206
[2,2,2] => 157
[2,2,1,1] => 338
[2,1,1,1,1] => 732
[1,1,1,1,1,1] => 1602
[7] => 1
[6,1] => 8
[5,2] => 27
[5,1,1] => 56
[4,3] => 49
[4,2,1] => 159
[4,1,1,1] => 332
[3,3,1] => 224
[3,2,2] => 353
[3,2,1,1] => 743
[3,1,1,1,1] => 1568
[2,2,2,1] => 1184
[2,2,1,1,1] => 2513
[2,1,1,1,1,1] => 5357
[1,1,1,1,1,1,1] => 11481
[8] => 1
[7,1] => 9
[6,2] => 35
[6,1,1] => 72
[5,3] => 76
[5,2,1] => 242
[5,1,1,1] => 500
[4,4] => 99
[4,3,1] => 436
[4,2,2] => 677
[4,2,1,1] => 1405
[4,1,1,1,1] => 2920
[3,3,2] => 943
[3,3,1,1] => 1965
[3,2,2,1] => 3078
[3,2,1,1,1] => 6437
[3,1,1,1,1,1] => 13487
[2,2,2,2] => 4845
[2,2,2,1,1] => 10163
[2,2,1,1,1,1] => 21378
[2,1,1,1,1,1,1] => 45104
[1,1,1,1,1,1,1,1] => 95503
[9] => 1
[8,1] => 10
[7,2] => 44
[7,1,1] => 90
[6,3] => 111
[6,2,1] => 349
[6,1,1,1] => 716
[5,4] => 175
[5,3,1] => 754
[5,2,2] => 1161
[5,2,1,1] => 2389
[5,1,1,1,1] => 4920
[4,4,1] => 972
[4,3,2] => 2059
[4,3,1,1] => 4249
[4,2,2,1] => 6577
[4,2,1,1,1] => 13599
[4,1,1,1,1,1] => 28147
[3,3,3] => 2836
[3,3,2,1] => 9095
[3,3,1,1,1] => 18843
[3,2,2,2] => 14139
[3,2,2,1,1] => 29338
[3,2,1,1,1,1] => 60962
[3,1,1,1,1,1,1] => 126866
[2,2,2,2,1] => 45796
[2,2,2,1,1,1] => 95359
[2,2,1,1,1,1,1] => 198920
[2,1,1,1,1,1,1,1] => 415781
[1,1,1,1,1,1,1,1,1] => 871030
[10] => 1
[9,1] => 11
[8,2] => 54
[8,1,1] => 110
[7,3] => 155
[7,2,1] => 483
[7,1,1,1] => 986
[6,4] => 286
[6,3,1] => 1214
[6,2,2] => 1859
[6,2,1,1] => 3803
[6,1,1,1,1] => 7784
[5,5] => 351
[5,4,1] => 1906
[5,3,2] => 3984
[5,3,1,1] => 8166
[5,2,2,1] => 12546
[5,2,1,1,1] => 25746
[5,1,1,1,1,1] => 52867
[4,4,2] => 5111
[4,4,1,1] => 10490
[4,3,3] => 6986
[4,3,2,1] => 22101
[4,3,1,1,1] => 45457
[4,2,2,2] => 34071
[4,2,2,1,1] => 70132
[4,2,1,1,1,1] => 144471
[4,1,1,1,1,1,1] => 297848
[3,3,3,1] => 30326
[3,3,2,2] => 46830
[3,3,2,1,1] => 96519
[3,3,1,1,1,1] => 199122
[3,2,2,2,1] => 149384
[3,2,2,1,1,1] => 308537
[3,2,1,1,1,1,1] => 637937
[3,1,1,1,1,1,1,1] => 1320510
[2,2,2,2,2] => 231571
[2,2,2,2,1,1] => 478940
[2,2,2,1,1,1,1] => 991743
[2,2,1,1,1,1,1,1] => 2056300
[2,1,1,1,1,1,1,1,1] => 4269680
[1,1,1,1,1,1,1,1,1,1] => 8879558
[11] => 1
[10,1] => 12
[9,2] => 65
[9,1,1] => 132
[8,3] => 209
[8,2,1] => 647
[8,1,1,1] => 1316
[7,4] => 441
[7,3,1] => 1852
[7,2,2] => 2825
[7,2,1,1] => 5755
[7,1,1,1,1] => 11728
[6,5] => 637
[6,4,1] => 3406
[6,3,2] => 7057
[6,3,1,1] => 14397
[6,2,2,1] => 22011
[6,2,1,1,1] => 44939
[6,1,1,1,1,1] => 91787
[5,5,1] => 4164
[5,4,2] => 11005
[5,4,1,1] => 22477
[5,3,3] => 14960
[5,3,2,1] => 46819
[5,3,1,1,1] => 95749
[5,2,2,2] => 71745
[5,2,2,1,1] => 146794
[5,2,1,1,1,1] => 300490
[5,1,1,1,1,1,1] => 615410
[4,4,3] => 19096
[4,4,2,1] => 59855
[4,4,1,1,1] => 122503
[4,3,3,1] => 81601
[4,3,2,2] => 125256
[4,3,2,1,1] => 256659
[4,3,1,1,1,1] => 526216
[4,2,2,2,1] => 394463
[4,2,2,1,1,1] => 809266
[4,2,1,1,1,1,1] => 1661295
[4,1,1,1,1,1,1,1] => 3412586
[3,3,3,2] => 171090
[3,3,3,1,1] => 350840
[3,3,2,2,1] => 539778
[3,3,2,1,1,1] => 1108391
[3,3,1,1,1,1,1] => 2277586
[3,2,2,2,2] => 831191
[3,2,2,2,1,1] => 1708151
[3,2,2,1,1,1,1] => 3512988
[3,2,1,1,1,1,1,1] => 7230549
[3,1,1,1,1,1,1,1,1] => 14894660
[2,2,2,2,2,1] => 2635257
[2,2,2,2,1,1,1] => 5424814
[2,2,2,1,1,1,1,1] => 11176952
[2,2,1,1,1,1,1,1,1] => 23049690
[2,1,1,1,1,1,1,1,1,1] => 47581727
[1,1,1,1,1,1,1,1,1,1,1] => 98329551
[12] => 1
[11,1] => 13
[10,2] => 77
[10,1,1] => 156
[9,3] => 274
[9,2,1] => 844
[9,1,1,1] => 1712
[8,4] => 650
[8,3,1] => 2708
[8,2,2] => 4119
[8,2,1,1] => 8365
[8,1,1,1,1] => 16992
[7,5] => 1078
[7,4,1] => 5699
[7,3,2] => 11734
[7,3,1,1] => 23856
[7,2,2,1] => 36346
[7,2,1,1,1] => 73932
[7,1,1,1,1,1] => 150427
[6,6] => 1275
[6,5,1] => 8213
[6,4,2] => 21483
[6,4,1,1] => 43716
[6,3,3] => 29094
[6,3,2,1] => 90344
[6,3,1,1,1] => 183999
[6,2,2,2] => 137868
[6,2,2,1,1] => 280874
[6,2,1,1,1,1] => 572393
[6,1,1,1,1,1,1] => 1166852
[5,5,2] => 26205
[5,5,1,1] => 53351
[5,4,3] => 45124
[5,4,2,1] => 140381
[5,4,1,1,1] => 286170
[5,3,3,1] => 190523
[5,3,2,2] => 291162
[5,3,2,1,1] => 593962
[5,3,1,1,1,1] => 1212100
[5,2,2,2,1] => 908387
[5,2,2,1,1,1] => 1854456
[5,2,1,1,1,1,1] => 3787285
[5,1,1,1,1,1,1,1] => 7737676
[4,4,4] => 57385
[4,4,3,1] => 242689
[4,4,2,2] => 371122
[4,4,2,1,1] => 757490
[4,4,1,1,1,1] => 1546726
[4,3,3,2] => 504587
[4,3,3,1,1] => 1030376
[4,3,2,2,1] => 1577875
[4,3,2,1,1,1] => 3224913
[4,3,1,1,1,1,1] => 6594182
[4,2,2,2,2] => 2417525
[4,2,2,2,1,1] => 4943420
[4,2,2,1,1,1,1] => 10113232
[4,2,1,1,1,1,1,1] => 20699788
[4,1,1,1,1,1,1,1,1] => 42390004
[3,3,3,3] => 686557
[3,3,3,2,1] => 2149402
[3,3,3,1,1,1] => 4395584
[3,3,2,2,2] => 3295560
[3,3,2,2,1,1] => 6743185
[3,3,2,1,1,1,1] => 13804634
[3,3,1,1,1,1,1,1] => 28276134
[3,2,2,2,2,1] => 10351471
[3,2,2,2,1,1,1] => 21204493
[3,2,2,1,1,1,1,1] => 43461404
[3,2,1,1,1,1,1,1,1] => 89133888
[3,1,1,1,1,1,1,1,1,1] => 182919233
[2,2,2,2,2,2] => 15902253
[2,2,2,2,2,1,1] => 32596683
[2,2,2,2,1,1,1,1] => 66858586
[2,2,2,1,1,1,1,1,1] => 137222522
[2,2,1,1,1,1,1,1,1,1] => 281835366
[2,1,1,1,1,1,1,1,1,1,1] => 579278088
[1,1,1,1,1,1,1,1,1,1,1,1] => 1191578522
-----------------------------------------------------------------------------
Created: May 20, 2017 at 17:37 by Martin Rubey
-----------------------------------------------------------------------------
Last Updated: May 20, 2017 at 22:45 by Martin Rubey