*****************************************************************************
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* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
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* This information is distributed in the hope that it will be useful, *
* but WITHOUT ANY WARRANTY; without even the implied warranty of *
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-----------------------------------------------------------------------------
Statistic identifier: St000812
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions.
For example, $h_{11} = 2m_{11} + m_2$, so the statistic on the partition $11$ is 3.
-----------------------------------------------------------------------------
References:
-----------------------------------------------------------------------------
Code:
def statistic(mu):
m = SymmetricFunctions(ZZ).m()
h = SymmetricFunctions(ZZ).h()
return sum(coeff for _, coeff in m(h(mu)))
-----------------------------------------------------------------------------
Statistic values:
[] => 1
[1] => 1
[2] => 2
[1,1] => 3
[3] => 3
[2,1] => 6
[1,1,1] => 10
[4] => 5
[3,1] => 12
[2,2] => 16
[2,1,1] => 27
[1,1,1,1] => 47
[5] => 7
[4,1] => 20
[3,2] => 32
[3,1,1] => 56
[2,2,1] => 76
[2,1,1,1] => 136
[1,1,1,1,1] => 246
[6] => 11
[5,1] => 35
[4,2] => 65
[4,1,1] => 114
[3,3] => 79
[3,2,1] => 191
[3,1,1,1] => 344
[2,2,2] => 263
[2,2,1,1] => 476
[2,1,1,1,1] => 870
[1,1,1,1,1,1] => 1602
[7] => 15
[6,1] => 54
[5,2] => 113
[5,1,1] => 202
[4,3] => 160
[4,2,1] => 398
[4,1,1,1] => 727
[3,3,1] => 493
[3,2,2] => 685
[3,2,1,1] => 1261
[3,1,1,1,1] => 2335
[2,2,2,1] => 1765
[2,2,1,1,1] => 3280
[2,1,1,1,1,1] => 6124
[1,1,1,1,1,1,1] => 11481
[8] => 22
[7,1] => 86
[6,2] => 199
[6,1,1] => 357
[5,3] => 318
[5,2,1] => 800
[5,1,1,1] => 1468
[4,4] => 371
[4,3,1] => 1167
[4,2,2] => 1632
[4,2,1,1] => 3021
[4,1,1,1,1] => 5623
[3,3,2] => 2041
[3,3,1,1] => 3790
[3,2,2,1] => 5342
[3,2,1,1,1] => 9997
[3,1,1,1,1,1] => 18782
[2,2,2,2] => 7548
[2,2,2,1,1] => 14162
[2,2,1,1,1,1] => 26673
[2,1,1,1,1,1,1] => 50399
[1,1,1,1,1,1,1,1] => 95503
[9] => 30
[8,1] => 128
[7,2] => 323
[7,1,1] => 585
[6,3] => 573
[6,2,1] => 1462
[6,1,1,1] => 2702
[5,4] => 756
[5,3,1] => 2435
[5,2,2] => 3423
[5,2,1,1] => 6387
[5,1,1,1,1] => 11966
[4,4,1] => 2872
[4,3,2] => 5090
[4,3,1,1] => 9537
[4,2,2,1] => 13520
[4,2,1,1,1] => 25487
[4,1,1,1,1,1] => 48192
[3,3,3] => 6417
[3,3,2,1] => 17109
[3,3,1,1,1] => 32313
[3,2,2,2] => 24339
[3,2,2,1,1] => 46068
[3,2,1,1,1,1] => 87431
[3,1,1,1,1,1,1] => 166334
[2,2,2,2,1] => 65786
[2,2,2,1,1,1] => 125088
[2,2,1,1,1,1,1] => 238388
[2,1,1,1,1,1,1,1] => 455249
[1,1,1,1,1,1,1,1,1] => 871030
[10] => 42
[9,1] => 192
[8,2] => 523
[8,1,1] => 951
[7,3] => 1013
[7,2,1] => 2607
[7,1,1,1] => 4836
[6,4] => 1485
[6,3,1] => 4849
[6,2,2] => 6844
[6,2,1,1] => 12820
[6,1,1,1,1] => 24103
[5,5] => 1683
[5,4,1] => 6539
[5,3,2] => 11694
[5,3,1,1] => 22018
[5,2,2,1] => 31338
[5,2,1,1,1] => 59315
[5,1,1,1,1,1] => 112565
[4,4,2] => 13905
[4,4,1,1] => 26220
[4,3,3] => 17617
[4,3,2,1] => 47494
[4,3,1,1,1] => 90161
[4,2,2,2] => 67880
[4,2,2,1,1] => 129091
[4,2,1,1,1,1] => 246064
[4,1,1,1,1,1,1] => 469990
[3,3,3,1] => 60442
[3,3,2,2] => 86494
[3,3,2,1,1] => 164735
[3,3,1,1,1,1] => 314432
[3,2,2,2,1] => 236463
[3,2,2,1,1,1] => 452025
[3,2,1,1,1,1,1] => 865691
[3,1,1,1,1,1,1,1] => 1660708
[2,2,2,2,2] => 339857
[2,2,2,2,1,1] => 650606
[2,2,2,1,1,1,1] => 1247675
[2,2,1,1,1,1,1,1] => 2396498
[2,1,1,1,1,1,1,1,1] => 4609878
[1,1,1,1,1,1,1,1,1,1] => 8879558
[11] => 56
[10,1] => 275
[9,2] => 803
[9,1,1] => 1470
[8,3] => 1683
[8,2,1] => 4376
[8,1,1,1] => 8159
[7,4] => 2701
[7,3,1] => 8956
[7,2,2] => 12689
[7,2,1,1] => 23890
[7,1,1,1,1] => 45113
[6,5] => 3405
[6,4,1] => 13519
[6,3,2] => 24382
[6,3,1,1] => 46158
[6,2,2,1] => 65943
[6,2,1,1,1] => 125379
[6,1,1,1,1,1] => 238901
[5,5,1] => 15469
[5,4,2] => 33373
[5,4,1,1] => 63336
[5,3,3] => 42473
[5,3,2,1] => 115700
[5,3,1,1,1] => 220766
[5,2,2,2] => 166038
[5,2,2,1,1] => 317293
[5,2,1,1,1,1] => 607439
[5,1,1,1,1,1,1] => 1164833
[4,4,3] => 50801
[4,4,2,1] => 138695
[4,4,1,1,1] => 264934
[4,3,3,1] => 177326
[4,3,2,2] => 254989
[4,3,2,1,1] => 488372
[4,3,1,1,1,1] => 936927
[4,2,2,2,1] => 704026
[4,2,2,1,1,1] => 1352328
[4,2,1,1,1,1,1] => 2601382
[4,1,1,1,1,1,1,1] => 5010794
[3,3,3,2] => 326656
[3,3,3,1,1] => 626353
[3,3,2,2,1] => 903777
[3,3,2,1,1,1] => 1737801
[3,3,1,1,1,1,1] => 3346113
[3,2,2,2,2] => 1305408
[3,2,2,2,1,1] => 2512916
[3,2,2,1,1,1,1] => 4843739
[3,2,1,1,1,1,1,1] => 9347861
[3,1,1,1,1,1,1,1,1] => 18060757
[2,2,2,2,2,1] => 3637109
[2,2,2,2,1,1,1] => 7017789
[2,2,2,1,1,1,1,1] => 13556488
[2,2,1,1,1,1,1,1,1] => 26215787
[2,1,1,1,1,1,1,1,1,1] => 50747824
[1,1,1,1,1,1,1,1,1,1,1] => 98329551
[12] => 77
[11,1] => 399
[10,2] => 1237
[10,1,1] => 2270
[9,3] => 2776
[9,2,1] => 7255
[9,1,1,1] => 13558
[8,4] => 4822
[8,3,1] => 16125
[8,2,2] => 22903
[8,2,1,1] => 43226
[8,1,1,1,1] => 81810
[7,5] => 6662
[7,4,1] => 26806
[7,3,2] => 48618
[7,3,1,1] => 92304
[7,2,2,1] => 132198
[7,2,1,1,1] => 251974
[7,1,1,1,1,1] => 481218
[6,6] => 7413
[6,5,1] => 34359
[6,4,2] => 74874
[6,4,1,1] => 142597
[6,3,3] => 95581
[6,3,2,1] => 262004
[6,3,1,1,1] => 501393
[6,2,2,2] => 377032
[6,2,2,1,1] => 722486
[6,2,1,1,1,1] => 1386715
[6,1,1,1,1,1,1] => 2665558
[5,5,2] => 86219
[5,5,1,1] => 164371
[5,4,3] => 132211
[5,4,2,1] => 363756
[5,4,1,1,1] => 697388
[5,3,3,1] => 466577
[5,3,2,2] => 673104
[5,3,2,1,1] => 1293446
[5,3,1,1,1,1] => 2489112
[5,2,2,2,1] => 1870045
[5,2,2,1,1,1] => 3602603
[5,2,1,1,1,1,1] => 6949104
[5,1,1,1,1,1,1,1] => 13419859
[4,4,4] => 158810
[4,4,3,1] => 561917
[4,4,2,2] => 811289
[4,4,2,1,1] => 1560329
[4,4,1,1,1,1] => 3005156
[4,3,3,2] => 1043121
[4,3,3,1,1] => 2008165
[4,3,2,2,1] => 2907772
[4,3,2,1,1,1] => 5611045
[4,3,1,1,1,1,1] => 10840145
[4,2,2,2,2] => 4214013
[4,2,2,2,1,1] => 8139459
[4,2,2,1,1,1,1] => 15739048
[4,2,1,1,1,1,1,1] => 30465586
[4,1,1,1,1,1,1,1,1] => 59028086
[3,3,3,3] => 1342215
[3,3,3,2,1] => 3747889
[3,3,3,1,1,1] => 7238283
[3,3,2,2,2] => 5435521
[3,3,2,2,1,1] => 10507313
[3,3,2,1,1,1,1] => 20333176
[3,3,1,1,1,1,1,1] => 39386682
[3,2,2,2,2,1] => 15264278
[3,2,2,2,1,1,1] => 29563182
[3,2,2,1,1,1,1,1] => 57310817
[3,2,1,1,1,1,1,1,1] => 111200335
[3,1,1,1,1,1,1,1,1,1] => 215941579
[2,2,2,2,2,2] => 22190989
[2,2,2,2,2,1,1] => 43012769
[2,2,2,2,1,1,1,1] => 83446864
[2,2,2,1,1,1,1,1,1] => 162027834
[2,2,1,1,1,1,1,1,1,1] => 314857712
[2,1,1,1,1,1,1,1,1,1,1] => 612300434
[1,1,1,1,1,1,1,1,1,1,1,1] => 1191578522
[13] => 101
[12,1] => 556
[10,3] => 4366
[9,2,2] => 39218
[8,5] => 12205
[8,4,1] => 49843
[8,3,2] => 90936
[8,3,1,1] => 173281
[7,6] => 14901
[7,5,1] => 70485
[7,4,2] => 155087
[7,3,3] => 198566
[6,6,1] => 79006
[6,5,2] => 201191
[6,5,1,1] => 385354
[6,4,3] => 310547
[6,4,2,1] => 860962
[6,3,2,2] => 1602689
[6,3,1,1,1,1] => 5965450
[6,2,2,1,1,1] => 8656230
[6,1,1,1,1,1,1,1] => 32417338
[5,5,3] => 359371
[5,4,4] => 433217
[5,4,3,1] => 1551710
[5,4,2,2] => 2248291
[5,4,2,1,1] => 4341108
[5,4,1,1,1,1] => 8391405
[5,3,3,2] => 2899935
[5,3,3,1,1] => 5603882
[5,3,2,2,1] => 8139289
[5,3,2,1,1,1] => 15758563
[5,3,1,1,1,1,1] => 30539286
[5,2,2,2,1,1] => 22923367
[5,2,2,1,1,1,1] => 44457921
[5,2,1,1,1,1,1,1] => 86295412
[4,4,4,1] => 1875822
[4,4,3,2] => 3510016
[4,4,3,1,1] => 6787502
[4,4,2,2,1] => 9864237
[4,3,3,3] => 4532834
[4,3,3,2,1] => 12756465
[3,3,3,3,1] => 16505959
[3,3,3,2,2] => 24033962
[3,3,2,2,2,1] => 68029106
[3,3,2,1,1,1,1,1] => 257335442
[3,2,2,2,2,2] => 99245511
[3,2,2,2,2,1,1] => 193076440
[2,2,2,2,2,2,1] => 282002663
[2,2,2,2,1,1,1,1,1] => 1070708800
-----------------------------------------------------------------------------
Created: May 20, 2017 at 17:54 by Martin Rubey
-----------------------------------------------------------------------------
Last Updated: Dec 29, 2023 at 11:12 by Martin Rubey