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-----------------------------------------------------------------------------
Statistic identifier: St000927
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The alternating sum of the coefficients of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1].
-----------------------------------------------------------------------------
References: [1] Garsia, A. M., Goupil, A. Character polynomials, their $q$-analogs and the Kronecker product [[MathSciNet:2576382]]
-----------------------------------------------------------------------------
Code:
def statistic(L):
return L.character_polynomial()(*[-1]*sum(L))
-----------------------------------------------------------------------------
Statistic values:
[2] => 1
[1,1] => 4
[3] => -1
[2,1] => -4
[1,1,1] => -7
[4] => 2
[3,1] => 4
[2,2] => 6
[2,1,1] => 9
[1,1,1,1] => 12
[5] => -2
[4,1] => -7
[3,2] => -10
[3,1,1] => -11
[2,2,1] => -17
[2,1,1,1] => -19
[1,1,1,1,1] => -19
[6] => 2
[5,1] => 9
[4,2] => 14
[4,1,1] => 22
[3,3] => 15
[3,2,1] => 36
[3,1,1,1] => 26
[2,2,2] => 15
[2,2,1,1] => 42
[2,1,1,1,1] => 34
[1,1,1,1,1,1] => 30
[7] => -2
[6,1] => -11
[5,2] => -22
[5,1,1] => -32
[4,3] => -29
[4,2,1] => -70
[4,1,1,1] => -53
[3,3,1] => -56
[3,2,2] => -48
[3,2,1,1] => -102
[3,1,1,1,1] => -53
[2,2,2,1] => -58
[2,2,1,1,1] => -83
[2,1,1,1,1,1] => -60
[1,1,1,1,1,1,1] => -45
[8] => 3
[7,1] => 12
[6,2] => 36
[6,1,1] => 41
[5,3] => 46
[5,2,1] => 125
[5,1,1,1] => 87
[4,4] => 39
[4,3,1] => 150
[4,2,2] => 127
[4,2,1,1] => 218
[4,1,1,1,1] => 116
[3,3,2] => 98
[3,3,1,1] => 174
[3,2,2,1] => 202
[3,2,1,1,1] => 233
[3,1,1,1,1,1] => 101
[2,2,2,2] => 68
[2,2,2,1,1] => 138
[2,2,1,1,1,1] => 158
[2,1,1,1,1,1,1] => 98
[1,1,1,1,1,1,1,1] => 67
[9] => -4
[8,1] => -15
[7,2] => -48
[7,1,1] => -52
[6,3] => -85
[6,2,1] => -202
[6,1,1,1] => -127
[5,4] => -84
[5,3,1] => -314
[5,2,2] => -260
[5,2,1,1] => -430
[5,1,1,1,1] => -206
[4,4,1] => -200
[4,3,2] => -371
[4,3,1,1] => -537
[4,2,2,1] => -552
[4,2,1,1,1] => -560
[4,1,1,1,1,1] => -231
[3,3,3] => -98
[3,3,2,1] => -472
[3,3,1,1,1] => -430
[3,2,2,2] => -274
[3,2,2,1,1] => -560
[3,2,1,1,1,1] => -484
[3,1,1,1,1,1,1] => -181
[2,2,2,2,1] => -212
[2,2,2,1,1,1] => -300
[2,2,1,1,1,1,1] => -275
[2,1,1,1,1,1,1,1] => -157
[1,1,1,1,1,1,1,1,1] => -97
[10] => 4
[9,1] => 20
[8,2] => 58
[8,1,1] => 71
[7,3] => 140
[7,2,1] => 300
[7,1,1,1] => 176
[6,4] => 157
[6,3,1] => 609
[6,2,2] => 447
[6,2,1,1] => 768
[6,1,1,1,1] => 326
[5,5] => 99
[5,4,1] => 594
[5,3,2] => 956
[5,3,1,1] => 1259
[5,2,2,1] => 1254
[5,2,1,1,1] => 1197
[5,1,1,1,1,1] => 446
[4,4,2] => 548
[4,4,1,1] => 775
[4,3,3] => 485
[4,3,2,1] => 1924
[4,3,1,1,1] => 1514
[4,2,2,2] => 803
[4,2,2,1,1] => 1696
[4,2,1,1,1,1] => 1256
[4,1,1,1,1,1,1] => 436
[3,3,3,1] => 554
[3,3,2,2] => 778
[3,3,2,1,1] => 1452
[3,3,1,1,1,1] => 960
[3,2,2,2,1] => 1041
[3,2,2,1,1,1] => 1322
[3,2,1,1,1,1,1] => 927
[3,1,1,1,1,1,1,1] => 309
[2,2,2,2,2] => 196
[2,2,2,2,1,1] => 539
[2,2,2,1,1,1,1] => 574
[2,2,1,1,1,1,1,1] => 465
[2,1,1,1,1,1,1,1,1] => 242
[1,1,1,1,1,1,1,1,1,1] => 139
[11] => -4
[10,1] => -24
[9,2] => -76
[9,1,1] => -94
[8,3] => -196
[8,2,1] => -428
[8,1,1,1] => -249
[7,4] => -299
[7,3,1] => -1056
[7,2,2] => -739
[7,2,1,1] => -1244
[7,1,1,1,1] => -497
[6,5] => -252
[6,4,1] => -1350
[6,3,2] => -2012
[6,3,1,1] => -2647
[6,2,2,1] => -2462
[6,2,1,1,1] => -2296
[6,1,1,1,1,1] => -765
[5,5,1] => -712
[5,4,2] => -2089
[5,4,1,1] => -2624
[5,3,3] => -1457
[5,3,2,1] => -5397
[5,3,1,1,1] => -3937
[5,2,2,2] => -2046
[5,2,2,1,1] => -4173
[5,2,1,1,1,1] => -2902
[5,1,1,1,1,1,1] => -895
[4,4,3] => -1026
[4,4,2,1] => -3240
[4,4,1,1,1] => -2338
[4,3,3,1] => -2970
[4,3,2,2] => -3503
[4,3,2,1,1] => -6576
[4,3,1,1,1,1] => -3711
[4,2,2,2,1] => -3530
[4,2,2,1,1,1] => -4292
[4,2,1,1,1,1,1] => -2598
[4,1,1,1,1,1,1,1] => -777
[3,3,3,2] => -1338
[3,3,3,1,1] => -1977
[3,3,2,2,1] => -3299
[3,3,2,1,1,1] => -3735
[3,3,1,1,1,1,1] => -1951
[3,2,2,2,2] => -1218
[3,2,2,2,1,1] => -2898
[3,2,2,1,1,1,1] => -2786
[3,2,1,1,1,1,1,1] => -1682
[3,1,1,1,1,1,1,1,1] => -509
[2,2,2,2,2,1] => -738
[2,2,2,2,1,1,1] => -1141
[2,2,2,1,1,1,1,1] => -1047
[2,2,1,1,1,1,1,1,1] => -751
[2,1,1,1,1,1,1,1,1,1] => -367
[1,1,1,1,1,1,1,1,1,1,1] => -195
[12] => 5
[11,1] => 27
[10,2] => 104
[10,1,1] => 115
[9,3] => 268
[9,2,1] => 596
[9,1,1,1] => 351
[8,4] => 507
[8,3,1] => 1670
[8,2,2] => 1195
[8,2,1,1] => 1904
[8,1,1,1,1] => 757
[7,5] => 531
[7,4,1] => 2715
[7,3,2] => 3782
[7,3,1,1] => 4979
[7,2,2,1] => 4434
[7,2,1,1,1] => 4033
[7,1,1,1,1,1] => 1250
[6,6] => 286
[6,5,1] => 2296
[6,4,2] => 5503
[6,4,1,1] => 6567
[6,3,3] => 3443
[6,3,2,1] => 12517
[6,3,1,1,1] => 8914
[6,2,2,2] => 4561
[6,2,2,1,1] => 8883
[6,2,1,1,1,1] => 5986
[6,1,1,1,1,1,1] => 1642
[5,5,2] => 2760
[5,5,1,1] => 3414
[5,4,3] => 4564
[5,4,2,1] => 13338
[5,4,1,1,1] => 8883
[5,3,3,1] => 9874
[5,3,2,2] => 10886
[5,3,2,1,1] => 20110
[5,3,1,1,1,1] => 10472
[5,2,2,2,1] => 9745
[5,2,2,1,1,1] => 11418
[5,2,1,1,1,1,1] => 6402
[5,1,1,1,1,1,1,1] => 1696
[4,4,4] => 1075
[4,4,3,1] => 7188
[4,4,2,2] => 6853
[4,4,2,1,1] => 12097
[4,4,1,1,1,1] => 6171
[4,3,3,2] => 7765
[4,3,3,1,1] => 11574
[4,3,2,2,1] => 16902
[4,3,2,1,1,1] => 18289
[4,3,1,1,1,1,1] => 8222
[4,2,2,2,2] => 4809
[4,2,2,2,1,1] => 10659
[4,2,2,1,1,1,1] => 9729
[4,2,1,1,1,1,1,1] => 5014
[4,1,1,1,1,1,1,1,1] => 1337
[3,3,3,3] => 1408
[3,3,3,2,1] => 6596
[3,3,3,1,1,1] => 5650
[3,3,2,2,2] => 4467
[3,3,2,2,1,1] => 10034
[3,3,2,1,1,1,1] => 8458
[3,3,1,1,1,1,1,1] => 3749
[3,2,2,2,2,1] => 4824
[3,2,2,2,1,1,1] => 6810
[3,2,2,1,1,1,1,1] => 5454
[3,2,1,1,1,1,1,1,1] => 2911
[3,1,1,1,1,1,1,1,1,1] => 814
[2,2,2,2,2,2] => 791
[2,2,2,2,2,1,1] => 1859
[2,2,2,2,1,1,1,1] => 2251
[2,2,2,1,1,1,1,1,1] => 1804
[2,2,1,1,1,1,1,1,1,1] => 1187
[2,1,1,1,1,1,1,1,1,1,1] => 541
[1,1,1,1,1,1,1,1,1,1,1,1] => 272
-----------------------------------------------------------------------------
Created: Aug 07, 2017 at 14:00 by Christian Stump
-----------------------------------------------------------------------------
Last Updated: Aug 07, 2017 at 14:00 by Christian Stump