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* 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: St000003
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The number of [[/StandardTableaux|standard Young tableaux]] of the partition.
-----------------------------------------------------------------------------
References:
-----------------------------------------------------------------------------
Code:
def statistic(x):
return StandardTableaux(x).cardinality()
-----------------------------------------------------------------------------
Statistic values:
[] => 1
[1] => 1
[2] => 1
[1,1] => 1
[3] => 1
[2,1] => 2
[1,1,1] => 1
[4] => 1
[3,1] => 3
[2,2] => 2
[2,1,1] => 3
[1,1,1,1] => 1
[5] => 1
[4,1] => 4
[3,2] => 5
[3,1,1] => 6
[2,2,1] => 5
[2,1,1,1] => 4
[1,1,1,1,1] => 1
[6] => 1
[5,1] => 5
[4,2] => 9
[4,1,1] => 10
[3,3] => 5
[3,2,1] => 16
[3,1,1,1] => 10
[2,2,2] => 5
[2,2,1,1] => 9
[2,1,1,1,1] => 5
[1,1,1,1,1,1] => 1
[7] => 1
[6,1] => 6
[5,2] => 14
[5,1,1] => 15
[4,3] => 14
[4,2,1] => 35
[4,1,1,1] => 20
[3,3,1] => 21
[3,2,2] => 21
[3,2,1,1] => 35
[3,1,1,1,1] => 15
[2,2,2,1] => 14
[2,2,1,1,1] => 14
[2,1,1,1,1,1] => 6
[1,1,1,1,1,1,1] => 1
[8] => 1
[7,1] => 7
[6,2] => 20
[6,1,1] => 21
[5,3] => 28
[5,2,1] => 64
[5,1,1,1] => 35
[4,4] => 14
[4,3,1] => 70
[4,2,2] => 56
[4,2,1,1] => 90
[4,1,1,1,1] => 35
[3,3,2] => 42
[3,3,1,1] => 56
[3,2,2,1] => 70
[3,2,1,1,1] => 64
[3,1,1,1,1,1] => 21
[2,2,2,2] => 14
[2,2,2,1,1] => 28
[2,2,1,1,1,1] => 20
[2,1,1,1,1,1,1] => 7
[1,1,1,1,1,1,1,1] => 1
[9] => 1
[8,1] => 8
[7,2] => 27
[7,1,1] => 28
[6,3] => 48
[6,2,1] => 105
[6,1,1,1] => 56
[5,4] => 42
[5,3,1] => 162
[5,2,2] => 120
[5,2,1,1] => 189
[5,1,1,1,1] => 70
[4,4,1] => 84
[4,3,2] => 168
[4,3,1,1] => 216
[4,2,2,1] => 216
[4,2,1,1,1] => 189
[4,1,1,1,1,1] => 56
[3,3,3] => 42
[3,3,2,1] => 168
[3,3,1,1,1] => 120
[3,2,2,2] => 84
[3,2,2,1,1] => 162
[3,2,1,1,1,1] => 105
[3,1,1,1,1,1,1] => 28
[2,2,2,2,1] => 42
[2,2,2,1,1,1] => 48
[2,2,1,1,1,1,1] => 27
[2,1,1,1,1,1,1,1] => 8
[1,1,1,1,1,1,1,1,1] => 1
[10] => 1
[9,1] => 9
[8,2] => 35
[8,1,1] => 36
[7,3] => 75
[7,2,1] => 160
[7,1,1,1] => 84
[6,4] => 90
[6,3,1] => 315
[6,2,2] => 225
[6,2,1,1] => 350
[6,1,1,1,1] => 126
[5,5] => 42
[5,4,1] => 288
[5,3,2] => 450
[5,3,1,1] => 567
[5,2,2,1] => 525
[5,2,1,1,1] => 448
[5,1,1,1,1,1] => 126
[4,4,2] => 252
[4,4,1,1] => 300
[4,3,3] => 210
[4,3,2,1] => 768
[4,3,1,1,1] => 525
[4,2,2,2] => 300
[4,2,2,1,1] => 567
[4,2,1,1,1,1] => 350
[4,1,1,1,1,1,1] => 84
[3,3,3,1] => 210
[3,3,2,2] => 252
[3,3,2,1,1] => 450
[3,3,1,1,1,1] => 225
[3,2,2,2,1] => 288
[3,2,2,1,1,1] => 315
[3,2,1,1,1,1,1] => 160
[3,1,1,1,1,1,1,1] => 36
[2,2,2,2,2] => 42
[2,2,2,2,1,1] => 90
[2,2,2,1,1,1,1] => 75
[2,2,1,1,1,1,1,1] => 35
[2,1,1,1,1,1,1,1,1] => 9
[1,1,1,1,1,1,1,1,1,1] => 1
[11] => 1
[10,1] => 10
[9,2] => 44
[9,1,1] => 45
[8,3] => 110
[8,2,1] => 231
[8,1,1,1] => 120
[7,4] => 165
[7,3,1] => 550
[7,2,2] => 385
[7,2,1,1] => 594
[7,1,1,1,1] => 210
[6,5] => 132
[6,4,1] => 693
[6,3,2] => 990
[6,3,1,1] => 1232
[6,2,2,1] => 1100
[6,2,1,1,1] => 924
[6,1,1,1,1,1] => 252
[5,5,1] => 330
[5,4,2] => 990
[5,4,1,1] => 1155
[5,3,3] => 660
[5,3,2,1] => 2310
[5,3,1,1,1] => 1540
[5,2,2,2] => 825
[5,2,2,1,1] => 1540
[5,2,1,1,1,1] => 924
[5,1,1,1,1,1,1] => 210
[4,4,3] => 462
[4,4,2,1] => 1320
[4,4,1,1,1] => 825
[4,3,3,1] => 1188
[4,3,2,2] => 1320
[4,3,2,1,1] => 2310
[4,3,1,1,1,1] => 1100
[4,2,2,2,1] => 1155
[4,2,2,1,1,1] => 1232
[4,2,1,1,1,1,1] => 594
[4,1,1,1,1,1,1,1] => 120
[3,3,3,2] => 462
[3,3,3,1,1] => 660
[3,3,2,2,1] => 990
[3,3,2,1,1,1] => 990
[3,3,1,1,1,1,1] => 385
[3,2,2,2,2] => 330
[3,2,2,2,1,1] => 693
[3,2,2,1,1,1,1] => 550
[3,2,1,1,1,1,1,1] => 231
[3,1,1,1,1,1,1,1,1] => 45
[2,2,2,2,2,1] => 132
[2,2,2,2,1,1,1] => 165
[2,2,2,1,1,1,1,1] => 110
[2,2,1,1,1,1,1,1,1] => 44
[2,1,1,1,1,1,1,1,1,1] => 10
[1,1,1,1,1,1,1,1,1,1,1] => 1
[12] => 1
[11,1] => 11
[10,2] => 54
[10,1,1] => 55
[9,3] => 154
[9,2,1] => 320
[9,1,1,1] => 165
[8,4] => 275
[8,3,1] => 891
[8,2,2] => 616
[8,2,1,1] => 945
[8,1,1,1,1] => 330
[7,5] => 297
[7,4,1] => 1408
[7,3,2] => 1925
[7,3,1,1] => 2376
[7,2,2,1] => 2079
[7,2,1,1,1] => 1728
[7,1,1,1,1,1] => 462
[6,6] => 132
[6,5,1] => 1155
[6,4,2] => 2673
[6,4,1,1] => 3080
[6,3,3] => 1650
[6,3,2,1] => 5632
[6,3,1,1,1] => 3696
[6,2,2,2] => 1925
[6,2,2,1,1] => 3564
[6,2,1,1,1,1] => 2100
[6,1,1,1,1,1,1] => 462
[5,5,2] => 1320
[5,5,1,1] => 1485
[5,4,3] => 2112
[5,4,2,1] => 5775
[5,4,1,1,1] => 3520
[5,3,3,1] => 4158
[5,3,2,2] => 4455
[5,3,2,1,1] => 7700
[5,3,1,1,1,1] => 3564
[5,2,2,2,1] => 3520
[5,2,2,1,1,1] => 3696
[5,2,1,1,1,1,1] => 1728
[5,1,1,1,1,1,1,1] => 330
[4,4,4] => 462
[4,4,3,1] => 2970
[4,4,2,2] => 2640
[4,4,2,1,1] => 4455
[4,4,1,1,1,1] => 1925
[4,3,3,2] => 2970
[4,3,3,1,1] => 4158
[4,3,2,2,1] => 5775
[4,3,2,1,1,1] => 5632
[4,3,1,1,1,1,1] => 2079
[4,2,2,2,2] => 1485
[4,2,2,2,1,1] => 3080
[4,2,2,1,1,1,1] => 2376
[4,2,1,1,1,1,1,1] => 945
[4,1,1,1,1,1,1,1,1] => 165
[3,3,3,3] => 462
[3,3,3,2,1] => 2112
[3,3,3,1,1,1] => 1650
[3,3,2,2,2] => 1320
[3,3,2,2,1,1] => 2673
[3,3,2,1,1,1,1] => 1925
[3,3,1,1,1,1,1,1] => 616
[3,2,2,2,2,1] => 1155
[3,2,2,2,1,1,1] => 1408
[3,2,2,1,1,1,1,1] => 891
[3,2,1,1,1,1,1,1,1] => 320
[3,1,1,1,1,1,1,1,1,1] => 55
[2,2,2,2,2,2] => 132
[2,2,2,2,2,1,1] => 297
[2,2,2,2,1,1,1,1] => 275
[2,2,2,1,1,1,1,1,1] => 154
[2,2,1,1,1,1,1,1,1,1] => 54
[2,1,1,1,1,1,1,1,1,1,1] => 11
[1,1,1,1,1,1,1,1,1,1,1,1] => 1
[8,5] => 572
[7,6] => 429
[7,5,1] => 2860
[7,4,2] => 6006
[6,6,1] => 1287
[5,5,3] => 3432
[5,4,4] => 2574
[5,4,3,1] => 15015
[5,4,2,2] => 12870
[5,4,2,1,1] => 21450
[5,4,1,1,1,1] => 9009
[5,3,3,2] => 11583
[5,3,3,1,1] => 16016
[5,3,2,2,1] => 21450
[5,3,2,1,1,1] => 20592
[4,4,4,1] => 3432
[4,4,3,2] => 8580
[4,4,3,1,1] => 11583
[4,4,2,2,1] => 12870
[4,3,3,3] => 3432
[4,3,3,2,1] => 15015
[3,3,3,3,1] => 2574
[3,3,3,2,2] => 3432
[3,3,2,2,2,1] => 5148
[3,2,2,2,2,2] => 1287
[2,2,2,2,2,2,1] => 429
[9,5] => 1001
[8,5,1] => 6006
[7,7] => 429
[7,5,2] => 14014
[7,4,3] => 16016
[6,6,2] => 6435
[6,4,4] => 9009
[6,2,2,2,2] => 14014
[5,5,4] => 6006
[5,5,1,1,1,1] => 14014
[5,4,3,2] => 48048
[5,4,3,1,1] => 64064
[5,4,2,2,1] => 68640
[5,4,2,1,1,1] => 63063
[5,3,3,2,1] => 64064
[5,3,2,2,2] => 35035
[5,2,2,2,2,1] => 21021
[4,4,4,2] => 12012
[4,4,3,3] => 12012
[4,4,3,2,1] => 48048
[4,3,2,2,2,1] => 36608
[3,3,3,3,2] => 6006
[3,3,3,3,1,1] => 9009
[3,3,2,2,2,2] => 6435
[2,2,2,2,2,2,2] => 429
[9,5,1] => 11375
[8,5,2] => 32032
[7,5,3] => 45045
[6,6,3] => 21450
[6,5,4] => 30030
[6,5,1,1,1,1] => 63700
[6,3,3,3] => 50050
[6,2,2,2,2,1] => 63700
[5,5,5] => 6006
[5,4,3,2,1] => 292864
[5,4,3,1,1,1] => 210210
[5,3,2,2,2,1] => 162162
[4,4,4,3] => 24024
[4,4,4,1,1,1] => 50050
[3,3,3,3,3] => 6006
[3,3,3,3,2,1] => 30030
[3,3,3,2,2,2] => 21450
[8,8] => 1430
[8,5,3] => 112112
[7,5,3,1] => 416988
[6,6,4] => 51480
[5,5,3,3] => 171600
[5,5,2,2,2] => 250250
[5,4,3,2,1,1] => 1153152
[5,4,2,2,2,1] => 630630
[4,4,4,4] => 24024
[4,4,4,2,2] => 171600
[4,3,3,3,2,1] => 292864
[3,3,3,3,2,2] => 51480
[2,2,2,2,2,2,2,2] => 1430
[8,6,3] => 272272
[6,5,3,3] => 972400
[6,5,2,2,2] => 1361360
[6,4,4,3] => 816816
[6,4,4,1,1,1] => 1591200
[6,3,3,3,2] => 1089088
[6,3,3,3,1,1] => 1591200
[5,5,4,3] => 583440
[5,5,4,1,1,1] => 1089088
[5,5,2,2,2,1] => 1361360
[5,4,3,2,2,1] => 3573570
[5,3,3,3,2,1] => 1633632
[4,4,4,3,2] => 583440
[4,4,4,3,1,1] => 816816
[4,4,4,2,2,1] => 972400
[4,4,4,3,2,1] => 3734016
[5,4,3,3,2,1] => 10720710
[6,3,3,3,2,1] => 6789120
[6,5,2,2,2,1] => 7920640
[5,5,3,3,1,1] => 6534528
[6,5,4,1,1,1] => 6789120
[5,5,3,3,2] => 4594590
[5,5,4,2,2] => 4594590
[6,4,4,2,2] => 6534528
[6,5,4,3] => 3734016
[2,2,2,2,2,2,2,2,2] => 4862
[3,3,3,3,3,3] => 87516
[6,6,6] => 87516
[9,6,3] => 678912
[8,6,4] => 787644
[9,9] => 4862
[5,4,4,3,2,1] => 34918884
[5,5,3,3,2,1] => 31039008
[5,5,4,2,2,1] => 29930472
[6,4,4,2,2,1] => 42325920
[5,5,4,3,1,1] => 26604864
[6,4,4,3,1,1] => 36732852
[6,5,3,3,1,1] => 42325920
[5,5,4,3,2] => 19399380
[6,4,4,3,2] => 26604864
[6,5,3,3,2] => 29930472
[6,5,4,2,2] => 31039008
[6,5,4,3,1] => 34918884
[6,5,4,1,1,1,1] => 23279256
[9,6,4] => 2116296
[8,5,4,2] => 16325712
[8,5,5,1] => 5038800
[5,5,4,3,2,1] => 141892608
[6,4,4,3,2,1] => 193489920
[6,5,3,3,2,1] => 214988800
[6,5,4,2,2,1] => 214988800
[6,5,4,3,1,1] => 193489920
[6,5,4,3,2] => 141892608
[6,5,2,2,2,2,1] => 81477396
[6,5,4,2,1,1,1] => 203693490
[7,5,4,3,1] => 154313250
[8,6,4,2] => 55099278
[10,6,4] => 5038800
[10,7,3] => 3779100
[9,7,4] => 5290740
[9,5,5,1] => 13856700
[6,5,4,3,2,1] => 1100742656
[6,3,3,3,3,2,1] => 203693490
[6,5,3,2,2,2,1] => 733296564
[6,5,4,3,1,1,1] => 814773960
[3,3,3,3,3,3,3] => 1385670
[11,7,3] => 8478750
[4,4,4,4,3,2,1] => 341429088
[6,4,3,3,3,2,1] => 2048574528
[9,6,4,3] => 611080470
[9,6,5,3] => 2007835830
[3,3,3,3,3,3,3,3] => 23371634
[11,7,5,1] => 1699755200
[8,8,8] => 23371634
-----------------------------------------------------------------------------
Created: Sep 14, 2011 at 16:53 by Chris Berg
-----------------------------------------------------------------------------
Last Updated: Sep 18, 2022 at 20:49 by Martin Rubey