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-----------------------------------------------------------------------------
Statistic identifier: St000712
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The number of semistandard Young tableau of given shape, with entries at most 4.
This is also the dimension of the corresponding irreducible representation of $GL_4$.
-----------------------------------------------------------------------------
References: [1] [[wikipedia:Young_tableau#Applications_in_representation_theory]]
-----------------------------------------------------------------------------
Code:
def statistic(mu):
return SemistandardTableaux(shape=mu, max_entry=4).cardinality()
-----------------------------------------------------------------------------
Statistic values:
[] => 1
[1] => 4
[2] => 10
[1,1] => 6
[3] => 20
[2,1] => 20
[1,1,1] => 4
[4] => 35
[3,1] => 45
[2,2] => 20
[2,1,1] => 15
[1,1,1,1] => 1
[5] => 56
[4,1] => 84
[3,2] => 60
[3,1,1] => 36
[2,2,1] => 20
[2,1,1,1] => 4
[1,1,1,1,1] => 0
[6] => 84
[5,1] => 140
[4,2] => 126
[4,1,1] => 70
[3,3] => 50
[3,2,1] => 64
[3,1,1,1] => 10
[2,2,2] => 10
[2,2,1,1] => 6
[2,1,1,1,1] => 0
[1,1,1,1,1,1] => 0
[7] => 120
[6,1] => 216
[5,2] => 224
[5,1,1] => 120
[4,3] => 140
[4,2,1] => 140
[4,1,1,1] => 20
[3,3,1] => 60
[3,2,2] => 36
[3,2,1,1] => 20
[3,1,1,1,1] => 0
[2,2,2,1] => 4
[2,2,1,1,1] => 0
[2,1,1,1,1,1] => 0
[1,1,1,1,1,1,1] => 0
[8] => 165
[7,1] => 315
[6,2] => 360
[6,1,1] => 189
[5,3] => 280
[5,2,1] => 256
[5,1,1,1] => 35
[4,4] => 105
[4,3,1] => 175
[4,2,2] => 84
[4,2,1,1] => 45
[4,1,1,1,1] => 0
[3,3,2] => 45
[3,3,1,1] => 20
[3,2,2,1] => 15
[3,2,1,1,1] => 0
[3,1,1,1,1,1] => 0
[2,2,2,2] => 1
[2,2,2,1,1] => 0
[2,2,1,1,1,1] => 0
[2,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1] => 0
[9] => 220
[8,1] => 440
[7,2] => 540
[7,1,1] => 280
[6,3] => 480
[6,2,1] => 420
[6,1,1,1] => 56
[5,4] => 280
[5,3,1] => 360
[5,2,2] => 160
[5,2,1,1] => 84
[5,1,1,1,1] => 0
[4,4,1] => 140
[4,3,2] => 140
[4,3,1,1] => 60
[4,2,2,1] => 36
[4,2,1,1,1] => 0
[4,1,1,1,1,1] => 0
[3,3,3] => 20
[3,3,2,1] => 20
[3,3,1,1,1] => 0
[3,2,2,2] => 4
[3,2,2,1,1] => 0
[3,2,1,1,1,1] => 0
[3,1,1,1,1,1,1] => 0
[2,2,2,2,1] => 0
[2,2,2,1,1,1] => 0
[2,2,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1] => 0
[10] => 286
[9,1] => 594
[8,2] => 770
[8,1,1] => 396
[7,3] => 750
[7,2,1] => 640
[7,1,1,1] => 84
[6,4] => 540
[6,3,1] => 630
[6,2,2] => 270
[6,2,1,1] => 140
[6,1,1,1,1] => 0
[5,5] => 196
[5,4,1] => 384
[5,3,2] => 300
[5,3,1,1] => 126
[5,2,2,1] => 70
[5,2,1,1,1] => 0
[5,1,1,1,1,1] => 0
[4,4,2] => 126
[4,4,1,1] => 50
[4,3,3] => 70
[4,3,2,1] => 64
[4,3,1,1,1] => 0
[4,2,2,2] => 10
[4,2,2,1,1] => 0
[4,2,1,1,1,1] => 0
[4,1,1,1,1,1,1] => 0
[3,3,3,1] => 10
[3,3,2,2] => 6
[3,3,2,1,1] => 0
[3,3,1,1,1,1] => 0
[3,2,2,2,1] => 0
[3,2,2,1,1,1] => 0
[3,2,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1] => 0
[2,2,2,2,2] => 0
[2,2,2,2,1,1] => 0
[2,2,2,1,1,1,1] => 0
[2,2,1,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1] => 0
[11] => 364
[10,1] => 780
[9,2] => 1056
[9,1,1] => 540
[8,3] => 1100
[8,2,1] => 924
[8,1,1,1] => 120
[7,4] => 900
[7,3,1] => 1000
[7,2,2] => 420
[7,2,1,1] => 216
[7,1,1,1,1] => 0
[6,5] => 504
[6,4,1] => 756
[6,3,2] => 540
[6,3,1,1] => 224
[6,2,2,1] => 120
[6,2,1,1,1] => 0
[6,1,1,1,1,1] => 0
[5,5,1] => 280
[5,4,2] => 360
[5,4,1,1] => 140
[5,3,3] => 160
[5,3,2,1] => 140
[5,3,1,1,1] => 0
[5,2,2,2] => 20
[5,2,2,1,1] => 0
[5,2,1,1,1,1] => 0
[5,1,1,1,1,1,1] => 0
[4,4,3] => 84
[4,4,2,1] => 60
[4,4,1,1,1] => 0
[4,3,3,1] => 36
[4,3,2,2] => 20
[4,3,2,1,1] => 0
[4,3,1,1,1,1] => 0
[4,2,2,2,1] => 0
[4,2,2,1,1,1] => 0
[4,2,1,1,1,1,1] => 0
[4,1,1,1,1,1,1,1] => 0
[3,3,3,2] => 4
[3,3,3,1,1] => 0
[3,3,2,2,1] => 0
[3,3,2,1,1,1] => 0
[3,3,1,1,1,1,1] => 0
[3,2,2,2,2] => 0
[3,2,2,2,1,1] => 0
[3,2,2,1,1,1,1] => 0
[3,2,1,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1,1] => 0
[2,2,2,2,2,1] => 0
[2,2,2,2,1,1,1] => 0
[2,2,2,1,1,1,1,1] => 0
[2,2,1,1,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1,1] => 0
[12] => 455
[11,1] => 1001
[10,2] => 1404
[10,1,1] => 715
[9,3] => 1540
[9,2,1] => 1280
[9,1,1,1] => 165
[8,4] => 1375
[8,3,1] => 1485
[8,2,2] => 616
[8,2,1,1] => 315
[8,1,1,1,1] => 0
[7,5] => 945
[7,4,1] => 1280
[7,3,2] => 875
[7,3,1,1] => 360
[7,2,2,1] => 189
[7,2,1,1,1] => 0
[7,1,1,1,1,1] => 0
[6,6] => 336
[6,5,1] => 735
[6,4,2] => 729
[6,4,1,1] => 280
[6,3,3] => 300
[6,3,2,1] => 256
[6,3,1,1,1] => 0
[6,2,2,2] => 35
[6,2,2,1,1] => 0
[6,2,1,1,1,1] => 0
[6,1,1,1,1,1,1] => 0
[5,5,2] => 280
[5,5,1,1] => 105
[5,4,3] => 256
[5,4,2,1] => 175
[5,4,1,1,1] => 0
[5,3,3,1] => 84
[5,3,2,2] => 45
[5,3,2,1,1] => 0
[5,3,1,1,1,1] => 0
[5,2,2,2,1] => 0
[5,2,2,1,1,1] => 0
[5,2,1,1,1,1,1] => 0
[5,1,1,1,1,1,1,1] => 0
[4,4,4] => 35
[4,4,3,1] => 45
[4,4,2,2] => 20
[4,4,2,1,1] => 0
[4,4,1,1,1,1] => 0
[4,3,3,2] => 15
[4,3,3,1,1] => 0
[4,3,2,2,1] => 0
[4,3,2,1,1,1] => 0
[4,3,1,1,1,1,1] => 0
[4,2,2,2,2] => 0
[4,2,2,2,1,1] => 0
[4,2,2,1,1,1,1] => 0
[4,2,1,1,1,1,1,1] => 0
[4,1,1,1,1,1,1,1,1] => 0
[3,3,3,3] => 1
[3,3,3,2,1] => 0
[3,3,3,1,1,1] => 0
[3,3,2,2,2] => 0
[3,3,2,2,1,1] => 0
[3,3,2,1,1,1,1] => 0
[3,3,1,1,1,1,1,1] => 0
[3,2,2,2,2,1] => 0
[3,2,2,2,1,1,1] => 0
[3,2,2,1,1,1,1,1] => 0
[3,2,1,1,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1,1,1] => 0
[2,2,2,2,2,2] => 0
[2,2,2,2,2,1,1] => 0
[2,2,2,2,1,1,1,1] => 0
[2,2,2,1,1,1,1,1,1] => 0
[2,2,1,1,1,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1,1,1] => 0
-----------------------------------------------------------------------------
Created: Mar 21, 2017 at 07:35 by Martin Rubey
-----------------------------------------------------------------------------
Last Updated: Jan 05, 2024 at 16:09 by Martin Rubey