*****************************************************************************
* 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: St001612
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition.
In particular, the value on the partition $(n)$ is the number of partitions of $n$, whereas the value on the partition $(1^n)$ is the number of permutations.
-----------------------------------------------------------------------------
References:
-----------------------------------------------------------------------------
Code:
def statistic(mu):
h = SymmetricFunctions(QQ).h()
F = species.PermutationSpecies().cycle_index_series()
return F.coefficient(mu.size()).scalar(h(mu))
-----------------------------------------------------------------------------
Statistic values:
[] => 1
[1] => 1
[2] => 2
[1,1] => 2
[3] => 3
[2,1] => 4
[1,1,1] => 6
[4] => 5
[3,1] => 7
[2,2] => 10
[2,1,1] => 14
[1,1,1,1] => 24
[5] => 7
[4,1] => 12
[3,2] => 18
[3,1,1] => 28
[2,2,1] => 38
[2,1,1,1] => 66
[1,1,1,1,1] => 120
[6] => 11
[5,1] => 19
[4,2] => 34
[4,1,1] => 52
[3,3] => 38
[3,2,1] => 84
[3,1,1,1] => 150
[2,2,2] => 120
[2,2,1,1] => 208
[2,1,1,1,1] => 384
[1,1,1,1,1,1] => 720
[7] => 15
[6,1] => 30
[5,2] => 56
[5,1,1] => 90
[4,3] => 74
[4,2,1] => 170
[4,1,1,1] => 306
[3,3,1] => 206
[3,2,2] => 290
[3,2,1,1] => 526
[3,1,1,1,1] => 984
[2,2,2,1] => 744
[2,2,1,1,1] => 1392
[2,1,1,1,1,1] => 2640
[1,1,1,1,1,1,1] => 5040
[8] => 22
[7,1] => 45
[6,2] => 94
[6,1,1] => 150
[5,3] => 133
[5,2,1] => 316
[5,1,1,1] => 576
[4,4] => 158
[4,3,1] => 450
[4,2,2] => 644
[4,2,1,1] => 1172
[4,1,1,1,1] => 2208
[3,3,2] => 788
[3,3,1,1] => 1464
[3,2,2,1] => 2086
[3,2,1,1,1] => 3954
[3,1,1,1,1,1] => 7560
[2,2,2,2] => 3000
[2,2,2,1,1] => 5664
[2,2,1,1,1,1] => 10848
[2,1,1,1,1,1,1] => 20880
[1,1,1,1,1,1,1,1] => 40320
[9] => 30
[8,1] => 67
[7,2] => 146
[7,1,1] => 240
[6,3] => 233
[6,2,1] => 560
[6,1,1,1] => 1026
[5,4] => 297
[5,3,1] => 899
[5,2,2] => 1284
[5,2,1,1] => 2380
[5,1,1,1,1] => 4512
[4,4,1] => 1058
[4,3,2] => 1886
[4,3,1,1] => 3536
[4,2,2,1] => 5074
[4,2,1,1,1] => 9678
[4,1,1,1,1,1] => 18600
[3,3,3] => 2370
[3,3,2,1] => 6424
[3,3,1,1,1] => 12300
[3,2,2,2] => 9264
[3,2,2,1,1] => 17744
[3,2,1,1,1,1] => 34224
[3,1,1,1,1,1,1] => 66240
[2,2,2,2,1] => 25656
[2,2,2,1,1,1] => 49536
[2,2,1,1,1,1,1] => 96000
[2,1,1,1,1,1,1,1] => 186480
[1,1,1,1,1,1,1,1,1] => 362880
[10] => 42
[9,1] => 97
[8,2] => 228
[8,1,1] => 374
[7,3] => 385
[7,2,1] => 946
[7,1,1,1] => 1746
[6,4] => 550
[6,3,1] => 1692
[6,2,2] => 2434
[6,2,1,1] => 4526
[6,1,1,1,1] => 8616
[5,5] => 602
[5,4,1] => 2254
[5,3,2] => 4074
[5,3,1,1] => 7714
[5,2,2,1] => 11118
[5,2,1,1,1] => 21330
[5,1,1,1,1,1] => 41160
[4,4,2] => 4868
[4,4,1,1] => 9188
[4,3,3] => 6146
[4,3,2,1] => 16920
[4,3,1,1,1] => 32586
[4,2,2,2] => 24552
[4,2,2,1,1] => 47248
[4,2,1,1,1,1] => 91536
[4,1,1,1,1,1,1] => 177840
[3,3,3,1] => 21642
[3,3,2,2] => 31380
[3,3,2,1,1] => 60636
[3,3,1,1,1,1] => 117648
[3,2,2,2,1] => 88152
[3,2,2,1,1,1] => 171216
[3,2,1,1,1,1,1] => 333360
[3,1,1,1,1,1,1,1] => 650160
[2,2,2,2,2] => 128400
[2,2,2,2,1,1] => 249456
[2,2,2,1,1,1,1] => 486144
[2,2,1,1,1,1,1,1] => 948960
[2,1,1,1,1,1,1,1,1] => 1854720
[1,1,1,1,1,1,1,1,1,1] => 3628800
[11] => 56
[10,1] => 139
[9,2] => 340
[9,1,1] => 568
[8,3] => 623
[8,2,1] => 1548
[8,1,1,1] => 2868
[7,4] => 951
[7,3,1] => 3023
[7,2,2] => 4348
[7,2,1,1] => 8164
[7,1,1,1,1] => 15600
[6,5] => 1166
[6,4,1] => 4496
[6,3,2] => 8210
[6,3,1,1] => 15624
[6,2,2,1] => 22604
[6,2,1,1,1] => 43524
[6,1,1,1,1,1] => 84240
[5,5,1] => 5110
[5,4,2] => 11214
[5,4,1,1] => 21410
[5,3,3] => 14302
[5,3,2,1] => 39826
[5,3,1,1,1] => 77058
[5,2,2,2] => 57936
[5,2,2,1,1] => 112144
[5,2,1,1,1,1] => 218016
[5,1,1,1,1,1,1] => 424800
[4,4,3] => 17170
[4,4,2,1] => 47896
[4,4,1,1,1] => 92736
[4,3,3,1] => 61628
[4,3,2,2] => 89786
[4,3,2,1,1] => 174310
[4,3,1,1,1,1] => 339528
[4,2,2,2,1] => 254448
[4,2,2,1,1,1] => 496032
[4,2,1,1,1,1,1] => 968880
[4,1,1,1,1,1,1,1] => 1895040
[3,3,3,2] => 115788
[3,3,3,1,1] => 225192
[3,3,2,2,1] => 328956
[3,3,2,1,1,1] => 641988
[3,3,1,1,1,1,1] => 1254960
[3,2,2,2,2] => 481032
[3,2,2,2,1,1] => 939408
[3,2,2,1,1,1,1] => 1837728
[3,2,1,1,1,1,1,1] => 3599280
[3,1,1,1,1,1,1,1,1] => 7056000
[2,2,2,2,2,1] => 1375680
[2,2,2,2,1,1,1] => 2692944
[2,2,2,1,1,1,1,1] => 5277600
[2,2,1,1,1,1,1,1,1] => 10352160
[2,1,1,1,1,1,1,1,1,1] => 20321280
[1,1,1,1,1,1,1,1,1,1,1] => 39916800
[12] => 77
[11,1] => 195
[10,2] => 506
[10,1,1] => 846
[9,3] => 977
[9,2,1] => 2456
[9,1,1,1] => 4572
[8,4] => 1614
[8,3,1] => 5194
[8,2,2] => 7504
[8,2,1,1] => 14128
[8,1,1,1,1] => 27072
[7,5] => 2133
[7,4,1] => 8470
[7,3,2] => 15592
[7,3,1,1] => 29834
[7,2,2,1] => 43280
[7,2,1,1,1] => 83616
[7,1,1,1,1,1] => 162240
[6,6] => 2382
[6,5,1] => 10772
[6,4,2] => 24008
[6,4,1,1] => 46026
[6,3,3] => 30740
[6,3,2,1] => 86264
[6,3,1,1,1] => 167454
[6,2,2,2] => 125904
[6,2,2,1,1] => 244400
[6,2,1,1,1,1] => 476352
[6,1,1,1,1,1,1] => 930240
[5,5,2] => 27556
[5,5,1,1] => 53040
[5,4,3] => 42696
[5,4,2,1] => 120346
[5,4,1,1,1] => 234024
[5,3,3,1] => 155582
[5,3,2,2] => 227388
[5,3,2,1,1] => 443164
[5,3,1,1,1,1] => 865776
[5,2,2,2,1] => 648816
[5,2,2,1,1,1] => 1268496
[5,2,1,1,1,1,1] => 2483760
[5,1,1,1,1,1,1,1] => 4868640
[4,4,4] => 51630
[4,4,3,1] => 188322
[4,4,2,2] => 275572
[4,4,2,1,1] => 537148
[4,4,1,1,1,1] => 1050000
[4,3,3,2] => 356996
[4,3,3,1,1] => 697068
[4,3,2,2,1] => 1021810
[4,3,2,1,1,1] => 2000478
[4,3,1,1,1,1,1] => 3921480
[4,2,2,2,2] => 1499208
[4,2,2,2,1,1] => 2936400
[4,2,2,1,1,1,1] => 5759616
[4,2,1,1,1,1,1,1] => 11307600
[4,1,1,1,1,1,1,1,1] => 22216320
[3,3,3,3] => 463176
[3,3,3,2,1] => 1327848
[3,3,3,1,1,1] => 2601540
[3,3,2,2,2] => 1948944
[3,3,2,2,1,1] => 3820704
[3,3,2,1,1,1,1] => 7498368
[3,3,1,1,1,1,1,1] => 14728320
[3,2,2,2,2,1] => 5614344
[3,2,2,2,1,1,1] => 11024352
[3,2,2,1,1,1,1,1] => 21664800
[3,2,1,1,1,1,1,1,1] => 42603120
[3,1,1,1,1,1,1,1,1,1] => 83825280
[2,2,2,2,2,2] => 8254800
[2,2,2,2,2,1,1] => 16216080
[2,2,2,2,1,1,1,1] => 31882176
[2,2,2,1,1,1,1,1,1] => 62722080
[2,2,1,1,1,1,1,1,1,1] => 123459840
[2,1,1,1,1,1,1,1,1,1,1] => 243129600
[1,1,1,1,1,1,1,1,1,1,1,1] => 479001600
[13] => 101
[12,1] => 272
[11,2] => 730
[11,1,1] => 1236
[10,3] => 1501
[10,2,1] => 3808
[10,1,1,1] => 7110
[9,4] => 2627
[9,3,1] => 8627
[9,2,2] => 12468
[9,2,1,1] => 23612
[9,1,1,1,1] => 45360
[8,5] => 3775
[8,4,1] => 15278
[8,3,2] => 28310
[8,3,1,1] => 54350
[8,2,2,1] => 79040
[8,2,1,1,1] => 153072
[8,1,1,1,1,1] => 297600
[7,6] => 4551
[7,5,1] => 21375
[7,4,2] => 48126
[7,4,1,1] => 92800
[7,3,3] => 61940
[7,3,2,1] => 174970
[7,3,1,1,1] => 340572
[7,2,2,2] => 255840
[7,2,2,1,1] => 498192
[7,2,1,1,1,1] => 973056
[7,1,1,1,1,1,1] => 1903680
[6,6,1] => 23926
[6,5,2] => 62378
[6,5,1,1] => 120610
[6,4,3] => 97468
[6,4,2,1] => 276644
[6,4,1,1,1] => 539556
[6,3,3,1] => 358850
[6,3,2,2] => 525854
[6,3,2,1,1] => 1027546
[6,3,1,1,1,1] => 2011944
[6,2,2,2,1] => 1507920
[6,2,2,1,1,1] => 2954400
[6,2,1,1,1,1,1] => 5795760
[6,1,1,1,1,1,1,1] => 11380320
[5,5,3] => 112966
[5,5,2,1] => 321288
[5,5,1,1,1] => 627168
[5,4,4] => 137070
[5,4,3,1] => 506946
[5,4,2,2] => 743732
[5,4,2,1,1] => 1455028
[5,4,1,1,1,1] => 2851872
[5,3,3,2] => 967368
[5,3,3,1,1] => 1894560
[5,3,2,2,1] => 2784342
[5,3,2,1,1,1] => 5464242
[5,3,1,1,1,1,1] => 10734120
[5,2,2,2,2] => 4094616
[5,2,2,2,1,1] => 8039520
[5,2,2,1,1,1,1] => 15801120
[5,2,1,1,1,1,1,1] => 31078800
[5,1,1,1,1,1,1,1,1] => 61165440
[4,4,4,1] => 616596
[4,4,3,2] => 1177924
[4,4,3,1,1] => 2307928
[4,4,2,2,1] => 3393488
[4,4,2,1,1,1] => 6662400
[4,4,1,1,1,1,1] => 13092960
[4,3,3,3] => 1534176
[4,3,3,2,1] => 4425532
[4,3,3,1,1,1] => 8693700
[4,3,2,2,2] => 6513648
[4,3,2,2,1,1] => 12801680
[4,3,2,1,1,1,1] => 25181376
[4,3,1,1,1,1,1,1] => 49564800
[4,2,2,2,2,1] => 18859152
[4,2,2,2,1,1,1] => 37112400
[4,2,2,1,1,1,1,1] => 73078080
[4,2,1,1,1,1,1,1,1] => 143972640
[4,1,1,1,1,1,1,1,1,1] => 283772160
[3,3,3,3,1] => 5774568
[3,3,3,2,2] => 8502540
[3,3,3,2,1,1] => 16719348
[3,3,3,1,1,1,1] => 32901264
[3,3,2,2,2,1] => 24639744
[3,3,2,2,1,1,1] => 48507552
[3,3,2,1,1,1,1,1] => 95549760
[3,3,1,1,1,1,1,1,1] => 188304480
[3,2,2,2,2,2] => 36326640
[3,2,2,2,2,1,1] => 71542176
[3,2,2,2,1,1,1,1] => 140973984
[3,2,2,1,1,1,1,1,1] => 277917120
[3,2,1,1,1,1,1,1,1,1] => 548110080
[3,1,1,1,1,1,1,1,1,1,1] => 1081382400
[2,2,2,2,2,2,1] => 105551280
[2,2,2,2,2,1,1,1] => 208059120
[2,2,2,2,1,1,1,1,1] => 410299200
[2,2,2,1,1,1,1,1,1,1] => 809434080
[2,2,1,1,1,1,1,1,1,1,1] => 1597397760
[14] => 135
[13,1] => 373
[12,2] => 1050
[12,1,1] => 1780
[11,3] => 2255
[11,2,1] => 5774
[11,1,1,1] => 10818
[10,4] => 4202
[10,3,1] => 13936
[10,2,2] => 20198
[10,2,1,1] => 38338
[10,1,1,1,1] => 73800
[9,5] => 6437
[9,4,1] => 26532
[9,3,2] => 49430
[9,3,1,1] => 95216
[9,2,2,1] => 138732
[9,2,1,1,1] => 269268
[9,1,1,1,1,1] => 524400
[8,6] => 8424
[8,5,1] => 40428
[8,4,2] => 91902
[8,4,1,1] => 177706
[8,3,3] => 118588
[8,3,2,1] => 336670
[8,3,1,1,1] => 656694
[8,2,2,2] => 493296
[8,2,2,1,1] => 962416
[8,2,1,1,1,1] => 1882944
[8,1,1,1,1,1,1] => 3689280
[7,7] => 9142
[7,6,1] => 49852
[7,5,2] => 131924
[7,5,1,1] => 256160
[7,4,3] => 207560
[7,4,2,1] => 592540
[7,4,1,1,1] => 1158528
[7,3,3,1] => 770730
[7,3,2,2] => 1131670
[7,3,2,1,1] => 2216250
[7,3,1,1,1,1] => 4347288
[7,2,2,2,1] => 3258336
[7,2,2,1,1,1] => 6395088
[7,2,1,1,1,1,1] => 12564720
[7,1,1,1,1,1,1,1] => 24706080
[6,6,2] => 148908
[6,6,1,1] => 289072
[6,5,3] => 272844
[6,5,2,1] => 780920
[6,5,1,1,1] => 1528554
[6,4,4] => 332306
[6,4,3,1] => 1239908
[6,4,2,2] => 1823414
[6,4,2,1,1] => 3575418
[6,4,1,1,1,1] => 7022040
[6,3,3,2] => 2377952
[6,3,3,1,1] => 4667352
[6,3,2,2,1] => 6873208
[6,3,2,1,1,1] => 13513224
[6,3,1,1,1,1,1] => 26589600
[6,2,2,2,2] => 10127304
[6,2,2,2,1,1] => 19918560
[6,2,2,1,1,1,1] => 39210240
[6,2,1,1,1,1,1,1] => 77233680
[6,1,1,1,1,1,1,1,1] => 152208000
[5,5,4] => 387146
[5,5,3,1] => 1448146
[5,5,2,2] => 2130104
[5,5,2,1,1] => 4179800
[5,5,1,1,1,1] => 8212416
[5,4,4,1] => 1767558
[5,4,3,2] => 3396002
[5,4,3,1,1] => 6671408
[5,4,2,2,1] => 9831618
[5,4,2,1,1,1] => 19343598
[5,4,1,1,1,1,1] => 38086440
[5,3,3,3] => 4436310
[5,3,3,2,1] => 12856144
[5,3,3,1,1,1] => 25305864
[5,3,2,2,2] => 18961344
[5,3,2,2,1,1] => 37338368
[5,3,2,1,1,1,1] => 73573584
[5,3,1,1,1,1,1,1] => 145048320
[5,2,2,2,2,1] => 55111848
[5,2,2,2,1,1,1] => 108634320
[5,2,2,1,1,1,1,1] => 214241280
[5,2,1,1,1,1,1,1,1] => 422689680
[5,1,1,1,1,1,1,1,1,1] => 834261120
[4,4,4,2] => 4151112
[4,4,4,1,1] => 8156976
[4,4,3,3] => 5424220
[4,4,3,2,1] => 15730404
[4,4,3,1,1,1] => 30973584
[4,4,2,2,2] => 23209152
[4,4,2,2,1,1] => 45715136
[4,4,2,1,1,1,1] => 90105312
[4,4,1,1,1,1,1,1] => 177687360
[4,3,3,3,1] => 20585340
[4,3,3,2,2] => 30380928
[4,3,3,2,1,1] => 59867472
[4,3,3,1,1,1,1] => 118038816
[4,3,2,2,2,1] => 88417584
[4,3,2,2,1,1,1] => 174387744
[4,3,2,1,1,1,1,1] => 344099520
[4,3,1,1,1,1,1,1,1] => 679230720
[4,2,2,2,2,2] => 130625040
[4,2,2,2,2,1,1] => 257710080
[4,2,2,2,1,1,1,1] => 508657824
[4,2,2,1,1,1,1,1,1] => 1004330880
[4,2,1,1,1,1,1,1,1,1] => 1983663360
[3,3,3,3,2] => 39784752
[3,3,3,3,1,1] => 78426528
[3,3,3,2,2,1] => 115860468
[3,3,3,2,1,1,1] => 228581964
[3,3,3,1,1,1,1,1] => 451155600
[3,3,2,2,2,2] => 171212304
[3,3,2,2,2,1,1] => 337886496
[3,3,2,2,1,1,1,1] => 667077696
[3,3,2,1,1,1,1,1,1] => 1317437280
[3,2,2,2,2,2,1] => 499588800
[3,2,2,2,2,1,1,1] => 986581584
[3,2,2,2,1,1,1,1,1] => 1948920480
[2,2,2,2,2,2,2] => 738864000
[2,2,2,2,2,2,1,1] => 1459457280
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Created: Sep 27, 2020 at 18:40 by Martin Rubey
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Last Updated: Sep 27, 2020 at 18:40 by Martin Rubey