*****************************************************************************
* www.FindStat.org - The Combinatorial Statistic Finder *
* *
* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
* *
* This information is distributed in the hope that it will be useful, *
* but WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. *
*****************************************************************************
-----------------------------------------------------------------------------
Statistic identifier: St001602
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions.
-----------------------------------------------------------------------------
References:
-----------------------------------------------------------------------------
Code:
def statistic(mu):
s = SymmetricFunctions(QQ).s()
A = CombinatorialSpecies()
X = species.SingletonSpecies()
E = species.SetSpecies()
A.define(X*E(A))
F = species.PermutationSpecies()(A).cycle_index_series()
return F.coefficient(mu.size()).scalar(s(mu))
-----------------------------------------------------------------------------
Statistic values:
[1] => 1
[2] => 3
[1,1] => 1
[3] => 7
[2,1] => 8
[1,1,1] => 4
[4] => 19
[3,1] => 33
[2,2] => 24
[2,1,1] => 27
[1,1,1,1] => 9
[5] => 47
[4,1] => 128
[3,2] => 141
[3,1,1] => 151
[2,2,1] => 119
[2,1,1,1] => 85
[1,1,1,1,1] => 20
[6] => 130
[5,1] => 441
[4,2] => 699
[4,1,1] => 695
[3,3] => 344
[3,2,1] => 1016
[3,1,1,1] => 587
[2,2,2] => 309
[2,2,1,1] => 486
[2,1,1,1,1] => 254
[1,1,1,1,1,1] => 45
[7] => 343
[6,1] => 1495
[5,2] => 2952
[5,1,1] => 2931
[4,3] => 2674
[4,2,1] => 6113
[4,1,1,1] => 3216
[3,3,1] => 3455
[3,2,2] => 3318
[3,2,1,1] => 5216
[3,1,1,1,1] => 2058
[2,2,2,1] => 1963
[2,2,1,1,1] => 1830
[2,1,1,1,1,1] => 725
[1,1,1,1,1,1,1] => 111
[8] => 951
[7,1] => 4883
[6,2] => 11756
[6,1,1] => 11530
[5,3] => 14623
[5,2,1] => 30987
[5,1,1,1] => 15764
[4,4] => 7017
[4,3,1] => 31419
[4,2,2] => 24189
[4,2,1,1] => 36892
[4,1,1,1,1] => 13285
[3,3,2] => 17176
[3,3,1,1] => 22029
[3,2,2,1] => 26440
[3,2,1,1,1] => 22787
[3,1,1,1,1,1] => 6874
[2,2,2,2] => 4976
[2,2,2,1,1] => 9408
[2,2,1,1,1,1] => 6345
[2,1,1,1,1,1,1] => 2039
[1,1,1,1,1,1,1,1] => 276
[9] => 2615
[8,1] => 15748
[7,2] => 44317
[7,1,1] => 43508
[6,3] => 69637
[6,2,1] => 142113
[6,1,1,1] => 70942
[5,4] => 56799
[5,3,1] => 200488
[5,2,2] => 142907
[5,2,1,1] => 215676
[5,1,1,1,1] => 74188
[4,4,1] => 99619
[4,3,2] => 187636
[4,3,1,1] => 231863
[4,2,2,1] => 223852
[4,2,1,1,1] => 185526
[4,1,1,1,1,1] => 50814
[3,3,3] => 44148
[3,3,2,1] => 167380
[3,3,1,1,1] => 113503
[3,2,2,2] => 79703
[3,2,2,1,1] => 148266
[3,2,1,1,1,1] => 90369
[3,1,1,1,1,1,1] => 22238
[2,2,2,2,1] => 36103
[2,2,2,1,1,1] => 39407
[2,2,1,1,1,1,1] => 20783
[2,1,1,1,1,1,1,1] => 5702
[1,1,1,1,1,1,1,1,1] => 666
[10] => 7318
[9,1] => 50054
[8,2] => 161848
[8,1,1] => 158530
[7,3] => 303642
[7,2,1] => 609037
[7,1,1,1] => 301251
[6,4] => 335433
[6,3,1] => 1084007
[6,2,2] => 747377
[6,2,1,1] => 1118689
[6,1,1,1,1] => 376160
[5,5] => 149827
[5,4,1] => 934739
[5,3,2] => 1383719
[5,3,1,1] => 1682891
[5,2,2,1] => 1507586
[5,2,1,1,1] => 1224236
[5,1,1,1,1,1] => 319364
[4,4,2] => 747207
[4,4,1,1] => 857886
[4,3,3] => 600389
[4,3,2,1] => 2090631
[4,3,1,1,1] => 1362268
[4,2,2,2] => 779984
[4,2,2,1,1] => 1426827
[4,2,1,1,1,1] => 831473
[4,1,1,1,1,1,1] => 184396
[3,3,3,1] => 543502
[3,3,2,2] => 632355
[3,3,2,1,1] => 1095900
[3,3,1,1,1,1] => 517200
[3,2,2,2,1] => 671697
[3,2,2,1,1,1] => 704295
[3,2,1,1,1,1,1] => 335763
[3,1,1,1,1,1,1,1] => 69889
[2,2,2,2,2] => 91548
[2,2,2,2,1,1] => 190161
[2,2,2,1,1,1,1] => 151099
[2,2,1,1,1,1,1,1] => 65952
[2,1,1,1,1,1,1,1,1] => 15782
[1,1,1,1,1,1,1,1,1,1] => 1618
[11] => 20491
[10,1] => 157780
[9,2] => 575068
[9,1,1] => 563741
[8,3] => 1251488
[8,2,1] => 2484790
[8,1,1,1] => 1222372
[7,4] => 1707296
[7,3,1] => 5299381
[7,2,2] => 3588664
[7,2,1,1] => 5349917
[7,1,1,1,1] => 1774629
[6,5] => 1291703
[6,4,1] => 6218244
[6,3,2] => 8456839
[6,3,1,1] => 10187557
[6,2,2,1] => 8822458
[6,2,1,1,1] => 7079072
[6,1,1,1,1,1] => 1798803
[5,5,1] => 2858639
[5,4,2] => 8036094
[5,4,1,1] => 9082522
[5,3,3] => 5188260
[5,3,2,1] => 17352513
[5,3,1,1,1] => 11067920
[5,2,2,2] => 5933784
[5,2,2,1,1] => 10757584
[5,2,1,1,1,1] => 6112049
[5,1,1,1,1,1,1] => 1287996
[4,4,3] => 3515188
[4,4,2,1] => 9601456
[4,4,1,1,1] => 5744585
[4,3,3,1] => 8383548
[4,3,2,2] => 9057507
[4,3,2,1,1] => 15409122
[4,3,1,1,1,1] => 6955992
[4,2,2,2,1] => 7398235
[4,2,2,1,1,1] => 7586236
[4,2,1,1,1,1,1] => 3441883
[4,1,1,1,1,1,1,1] => 643234
[3,3,3,2] => 3030590
[3,3,3,1,1] => 4213456
[3,3,2,2,1] => 6156194
[3,3,2,1,1,1] => 5922497
[3,3,1,1,1,1,1] => 2168628
[3,2,2,2,2] => 1949331
[3,2,2,2,1,1] => 3990844
[3,2,2,1,1,1,1] => 3021837
[3,2,1,1,1,1,1,1] => 1189785
[3,1,1,1,1,1,1,1,1] => 214585
[2,2,2,2,2,1] => 714378
[2,2,2,2,1,1,1] => 862836
[2,2,2,1,1,1,1,1] => 546107
[2,2,1,1,1,1,1,1,1] => 204611
[2,1,1,1,1,1,1,1,1,1] => 43259
[1,1,1,1,1,1,1,1,1,1,1] => 4017
[12] => 57903
[11,1] => 493441
[10,2] => 2003665
[10,1,1] => 1964179
[9,3] => 4945437
[9,2,1] => 9764801
[9,1,1,1] => 4790326
[8,4] => 7960553
[8,3,1] => 24166972
[8,2,2] => 16203849
[8,2,1,1] => 24087817
[8,1,1,1,1] => 7928086
[7,5] => 8032446
[7,4,1] => 35192705
[7,3,2] => 45986266
[7,3,1,1] => 55094809
[7,2,2,1] => 46853100
[7,2,1,1,1] => 37323659
[7,1,1,1,1,1] => 9334812
[6,6] => 3454759
[6,5,1] => 27509562
[6,4,2] => 59982316
[6,4,1,1] => 67132743
[6,3,3] => 35953378
[6,3,2,1] => 117669543
[6,3,1,1,1] => 74117760
[6,2,2,2] => 38619079
[6,2,2,1,1] => 69591781
[6,2,1,1,1,1] => 38956533
[6,1,1,1,1,1,1] => 7973068
[5,5,2] => 28714329
[5,5,1,1] => 31397152
[5,4,3] => 43981477
[5,4,2,1] => 115386199
[5,4,1,1,1] => 67554399
[5,3,3,1] => 80799343
[5,3,2,2] => 84325120
[5,3,2,1,1] => 141990915
[5,3,1,1,1,1] => 62507326
[5,2,2,2,1] => 62478917
[5,2,2,1,1,1] => 63223877
[5,2,1,1,1,1,1] => 27890959
[5,1,1,1,1,1,1,1] => 4938240
[4,4,4] => 9210566
[4,4,3,1] => 56069180
[4,4,2,2] => 48565468
[4,4,2,1,1] => 79855194
[4,4,1,1,1,1] => 32840479
[4,3,3,2] => 53172292
[4,3,3,1,1] => 72585160
[4,3,2,2,1] => 98351648
[4,3,2,1,1,1] => 92505243
[4,3,1,1,1,1,1] => 32248843
[4,2,2,2,2] => 24093179
[4,2,2,2,1,1] => 48798953
[4,2,2,1,1,1,1] => 35997249
[4,2,1,1,1,1,1,1] => 13444088
[4,1,1,1,1,1,1,1,1] => 2173367
[3,3,3,3] => 7844047
[3,3,3,2,1] => 34570311
[3,3,3,1,1,1] => 26068121
[3,3,2,2,2] => 20851166
[3,3,2,2,1,1] => 41261745
[3,3,2,1,1,1,1] => 28424977
[3,3,1,1,1,1,1,1] => 8548160
[3,2,2,2,2,1] => 17027562
[3,2,2,2,1,1,1] => 20095913
[3,2,2,1,1,1,1,1] => 12088852
[3,2,1,1,1,1,1,1,1] => 4066989
[3,1,1,1,1,1,1,1,1,1] => 647678
[2,2,2,2,2,2] => 1820231
[2,2,2,2,2,1,1] => 4007963
[2,2,2,2,1,1,1,1] => 3569980
[2,2,2,1,1,1,1,1,1] => 1893192
[2,2,1,1,1,1,1,1,1,1] => 622202
[2,1,1,1,1,1,1,1,1,1,1] => 117912
[1,1,1,1,1,1,1,1,1,1,1,1] => 10006
[13] => 163898
[12,1] => 1534884
[11,2] => 6866042
[11,1,1] => 6735856
[10,3] => 18932370
[10,2,1] => 37257908
[10,1,1,1] => 18248596
[9,4] => 34926525
[9,3,1] => 104689964
[9,2,2] => 69765576
[9,2,1,1] => 103548408
[9,1,1,1,1] => 33915591
[8,5] => 43083435
[8,4,1] => 180311081
[8,3,2] => 230557039
[8,3,1,1] => 275262177
[8,2,2,1] => 231632829
[8,2,1,1,1] => 183679631
[8,1,1,1,1,1] => 45491926
[7,6] => 30862502
[7,5,1] => 188805866
[7,4,2] => 375258494
[7,4,1,1] => 417527639
[7,3,3] => 217400912
[7,3,2,1] => 702439122
[7,3,1,1,1] => 439019063
[7,2,2,2] => 225396305
[7,2,2,1,1] => 404637998
[7,2,1,1,1,1] => 224393866
[7,1,1,1,1,1,1] => 45118576
[6,6,1] => 82507937
[6,5,2] => 308145050
[6,5,1,1] => 333478433
[6,4,3] => 369974178
[6,4,2,1] => 949445830
[6,4,1,1,1] => 548727730
[6,3,3,1] => 616847240
[6,3,2,2] => 631958558
[6,3,2,1,1] => 1057274716
[6,3,1,1,1,1] => 458458454
[6,2,2,2,1] => 446236187
[6,2,2,1,1,1] => 447781496
[6,2,1,1,1,1,1] => 194186707
[6,1,1,1,1,1,1,1] => 33323070
[5,5,3] => 191912104
[5,5,2,1] => 461889786
[5,5,1,1,1] => 259632180
[5,4,4] => 140170118
[5,4,3,1] => 777627957
[5,4,2,2] => 650540673
[5,4,2,1,1] => 1058536010
[5,4,1,1,1,1] => 424405370
[5,3,3,2] => 571013146
[5,3,3,1,1] => 771132672
[5,3,2,2,1] => 1009017695
[5,3,2,1,1,1] => 936261132
[5,3,1,1,1,1,1] => 317412076
[5,2,2,2,2] => 224884082
[5,2,2,2,1,1] => 452474815
[5,2,2,1,1,1,1] => 328469438
[5,2,1,1,1,1,1,1] => 119017787
[5,1,1,1,1,1,1,1,1] => 18187641
[4,4,4,1] => 170939928
[4,4,3,2] => 412137147
[4,4,3,1,1] => 543527557
[4,4,2,2,1] => 590137101
[4,4,2,1,1,1] => 532551999
[4,4,1,1,1,1,1] => 168209289
[4,3,3,3] => 158953807
[4,3,3,2,1] => 672100179
[4,3,3,1,1,1] => 495345910
[4,3,2,2,2] => 371401367
[4,3,2,2,1,1] => 726807599
[4,3,2,1,1,1,1] => 487819474
[4,3,1,1,1,1,1,1] => 139227587
[4,2,2,2,2,1] => 231939400
[4,2,2,2,1,1,1] => 269892707
[4,2,2,1,1,1,1,1] => 157718591
[4,2,1,1,1,1,1,1,1] => 50214314
[4,1,1,1,1,1,1,1,1,1] => 7156167
[3,3,3,3,1] => 109855195
[3,3,3,2,2] => 143320388
[3,3,3,2,1,1] => 263057036
[3,3,3,1,1,1,1] => 140222034
[3,3,2,2,2,1] => 203842565
[3,3,2,2,1,1,1] => 230689740
[3,3,2,1,1,1,1,1] => 125590365
[3,3,1,1,1,1,1,1,1] => 32136763
[3,2,2,2,2,2] => 48368371
[3,2,2,2,2,1,1] => 105389293
[3,2,2,2,1,1,1,1] => 91340349
[3,2,2,1,1,1,1,1,1] => 45923314
[3,2,1,1,1,1,1,1,1,1] => 13512735
[3,1,1,1,1,1,1,1,1,1,1] => 1928348
[2,2,2,2,2,2,1] => 14871610
[2,2,2,2,2,1,1,1] => 19295410
[2,2,2,2,1,1,1,1,1] => 13863944
[2,2,2,1,1,1,1,1,1,1] => 6357081
[2,2,1,1,1,1,1,1,1,1,1] => 1861237
[2,1,1,1,1,1,1,1,1,1,1,1] => 320098
[1,1,1,1,1,1,1,1,1,1,1,1,1] => 24844
[14] => 466199
[13,1] => 4751067
[12,2] => 23220889
[12,1,1] => 22792381
[11,3] => 70660963
[11,2,1] => 138798812
[11,1,1,1] => 67934060
[10,4] => 146533592
[10,3,1] => 435779847
[10,2,2] => 289312657
[10,2,1,1] => 428975159
[10,1,1,1,1] => 140094715
[9,5] => 211206424
[9,4,1] => 861831551
[9,3,2] => 1087926899
[9,3,1,1] => 1296040948
[9,2,2,1] => 1083506723
[9,2,1,1,1] => 856696497
[9,1,1,1,1,1] => 210846304
[8,6] => 199590706
[8,5,1] => 1106181430
[8,4,2] => 2102116342
[8,3,3] => 1195738351
[8,2,2,2] => 1213429834
[8,2,1,1,1,1] => 1197685752
[8,1,1,1,1,1,1] => 238088967
[7,7] => 83164062
[7,6,1] => 809114769
[7,2,1,1,1,1,1] => 1212899049
[7,1,1,1,1,1,1,1] => 204140465
[6,6,2] => 1040046726
[6,6,1,1] => 1105345887
[6,4,4] => 1351682301
[6,2,2,2,2] => 1755726690
[6,2,1,1,1,1,1,1] => 898485923
[6,1,1,1,1,1,1,1,1] => 132869758
[5,5,4] => 878598627
[5,5,1,1,1,1] => 1791894547
[5,3,3,3] => 1913295926
[5,3,1,1,1,1,1,1] => 1488995568
[5,2,2,1,1,1,1,1] => 1563455526
[5,2,1,1,1,1,1,1,1] => 482026626
[5,1,1,1,1,1,1,1,1,1] => 64842108
[4,4,4,2] => 1528488191
[4,4,4,1,1] => 1869834480
[4,4,3,3] => 1494808371
[4,4,1,1,1,1,1,1] => 794247984
[4,3,1,1,1,1,1,1,1] => 568895541
[4,2,2,2,2,2] => 724482057
[4,2,2,2,2,1,1] => 1567164433
[4,2,2,2,1,1,1,1] => 1335796407
[4,2,2,1,1,1,1,1,1] => 651000095
[4,2,1,1,1,1,1,1,1,1] => 180957588
[4,1,1,1,1,1,1,1,1,1,1] => 23072233
[3,3,3,3,2] => 657082503
[3,3,3,3,1,1] => 966400608
[3,3,3,2,2,1] => 1579308058
[3,3,3,2,1,1,1] => 1637308947
[3,3,3,1,1,1,1,1] => 684228704
[3,3,2,2,2,2] => 651027342
[3,3,2,2,2,1,1] => 1391798860
[3,3,2,2,1,1,1,1] => 1150806320
[3,3,2,1,1,1,1,1,1] => 521658141
[3,3,1,1,1,1,1,1,1,1] => 116403383
[3,2,2,2,2,2,1] => 433351822
[3,2,2,2,2,1,1,1] => 554077940
[3,2,2,2,1,1,1,1,1] => 386453490
[3,2,2,1,1,1,1,1,1,1] => 167590962
[3,2,1,1,1,1,1,1,1,1,1] => 43873472
[3,1,1,1,1,1,1,1,1,1,1,1] => 5673705
[2,2,2,2,2,2,2] => 38064059
[2,2,2,2,2,2,1,1] => 87325799
[2,2,2,2,2,1,1,1,1] => 84558244
[2,2,2,2,1,1,1,1,1,1] => 51392991
[2,2,2,1,1,1,1,1,1,1,1] => 20806755
[2,2,1,1,1,1,1,1,1,1,1,1] => 5497254
[2,1,1,1,1,1,1,1,1,1,1,1,1] => 865394
[1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 61924
[15] => 1328993
[14,1] => 14650472
[13,2] => 77656094
[13,1,1] => 76277462
[12,3] => 258353178
[12,2,1] => 506949725
[12,1,1,1] => 248060238
[11,4] => 593679559
[11,3,1] => 1756905162
[11,2,2] => 1163556871
[11,2,1,1] => 1724327614
[11,1,1,1,1] => 562130089
[10,5] => 972970999
[10,1,1,1,1,1] => 938066335
[9,6] => 1113418862
[9,1,1,1,1,1,1] => 1187996973
[8,7] => 764820279
[8,1,1,1,1,1,1,1] => 1159374758
[7,1,1,1,1,1,1,1,1] => 875914505
[6,1,1,1,1,1,1,1,1,1] => 509785347
[5,2,1,1,1,1,1,1,1,1] => 1871942391
[5,1,1,1,1,1,1,1,1,1,1] => 225027951
[4,2,1,1,1,1,1,1,1,1,1] => 633304011
[4,1,1,1,1,1,1,1,1,1,1,1] => 73088079
[3,3,3,3,3] => 1709646319
[3,3,2,1,1,1,1,1,1,1] => 2064913017
[3,3,1,1,1,1,1,1,1,1,1] => 409065407
[3,2,2,2,2,2,2] => 1214078549
[3,2,2,2,1,1,1,1,1,1] => 1549474924
[3,2,2,1,1,1,1,1,1,1,1] => 592236331
[3,2,1,1,1,1,1,1,1,1,1,1] => 139757331
[3,1,1,1,1,1,1,1,1,1,1,1,1] => 16525365
[2,2,2,2,2,2,2,1] => 321418259
[2,2,2,2,2,2,1,1,1] => 439916729
[2,2,2,2,2,1,1,1,1,1] => 347096922
[2,2,2,2,1,1,1,1,1,1,1] => 183750367
[2,2,2,1,1,1,1,1,1,1,1,1] => 66693596
[2,2,1,1,1,1,1,1,1,1,1,1,1] => 16066799
[2,1,1,1,1,1,1,1,1,1,1,1,1,1] => 2331217
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 155123
[16] => 3799624
[15,1] => 45025792
[14,2] => 257276727
[14,1,1] => 252865940
[13,3] => 928500885
[13,2,1] => 1821037070
[13,1,1,1] => 891128587
[8,8] => 2073523096
[6,1,1,1,1,1,1,1,1,1,1] => 1894170931
[5,1,1,1,1,1,1,1,1,1,1,1] => 763364397
[4,1,1,1,1,1,1,1,1,1,1,1,1] => 228069815
[3,3,1,1,1,1,1,1,1,1,1,1] => 1401797177
[3,2,2,1,1,1,1,1,1,1,1,1] => 2038283551
[3,2,1,1,1,1,1,1,1,1,1,1,1] => 438097447
[3,1,1,1,1,1,1,1,1,1,1,1,1,1] => 47724923
[2,2,2,2,2,2,2,2] => 825736331
[2,2,2,2,2,2,2,1,1] => 1954038143
[2,2,2,2,2,2,1,1,1,1] => 2016436391
[2,2,2,2,2,1,1,1,1,1,1] => 1356704514
[2,2,2,2,1,1,1,1,1,1,1,1] => 638107591
[2,2,2,1,1,1,1,1,1,1,1,1,1] => 210114264
[2,2,1,1,1,1,1,1,1,1,1,1,1,1] => 46531170
[2,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 6262169
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 389289
[17] => 10884049
[16,1] => 137993681
[15,2] => 845474608
[15,1,1] => 831521798
[4,1,1,1,1,1,1,1,1,1,1,1,1,1] => 702522095
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Created: Sep 26, 2020 at 23:55 by Martin Rubey
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Last Updated: Sep 26, 2020 at 23:55 by Martin Rubey