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* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
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-----------------------------------------------------------------------------
Statistic identifier: St000047
-----------------------------------------------------------------------------
Collection: Integer compositions
-----------------------------------------------------------------------------
Description: The number of standard immaculate tableaux of a given shape.
See Proposition 3.13 of [2] for a hook-length counting formula of these tableaux.
-----------------------------------------------------------------------------
References: [1] Berg, C., Bergeron, N., Saliola, F., Serrano, L., Zabrocki, M. The immaculate basis of the non-commutative symmetric functions [[arXiv:1303.4801]]
[2] Berg, C., Bergeron, N., Saliola, F., Serrano, L., Zabrocki, M. A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions [[arXiv:1208.5191]]
-----------------------------------------------------------------------------
Code:
def statistic(mu):
F = QuasiSymmetricFunctions(ZZ).F()
dI = QuasiSymmetricFunctions(ZZ).dI()
return sum(coeff for _, coeff in F(dI(mu)))
-----------------------------------------------------------------------------
Statistic values:
[1] => 1
[1,1] => 1
[2] => 1
[1,1,1] => 1
[1,2] => 1
[2,1] => 2
[3] => 1
[1,1,1,1] => 1
[1,1,2] => 1
[1,2,1] => 2
[1,3] => 1
[2,1,1] => 3
[2,2] => 3
[3,1] => 3
[4] => 1
[1,1,1,1,1] => 1
[1,1,1,2] => 1
[1,1,2,1] => 2
[1,1,3] => 1
[1,2,1,1] => 3
[1,2,2] => 3
[1,3,1] => 3
[1,4] => 1
[2,1,1,1] => 4
[2,1,2] => 4
[2,2,1] => 8
[2,3] => 4
[3,1,1] => 6
[3,2] => 6
[4,1] => 4
[5] => 1
[1,1,1,1,1,1] => 1
[1,1,1,1,2] => 1
[1,1,1,2,1] => 2
[1,1,1,3] => 1
[1,1,2,1,1] => 3
[1,1,2,2] => 3
[1,1,3,1] => 3
[1,1,4] => 1
[1,2,1,1,1] => 4
[1,2,1,2] => 4
[1,2,2,1] => 8
[1,2,3] => 4
[1,3,1,1] => 6
[1,3,2] => 6
[1,4,1] => 4
[1,5] => 1
[2,1,1,1,1] => 5
[2,1,1,2] => 5
[2,1,2,1] => 10
[2,1,3] => 5
[2,2,1,1] => 15
[2,2,2] => 15
[2,3,1] => 15
[2,4] => 5
[3,1,1,1] => 10
[3,1,2] => 10
[3,2,1] => 20
[3,3] => 10
[4,1,1] => 10
[4,2] => 10
[5,1] => 5
[6] => 1
[1,1,1,1,1,1,1] => 1
[1,1,1,1,1,2] => 1
[1,1,1,1,2,1] => 2
[1,1,1,1,3] => 1
[1,1,1,2,1,1] => 3
[1,1,1,2,2] => 3
[1,1,1,3,1] => 3
[1,1,1,4] => 1
[1,1,2,1,1,1] => 4
[1,1,2,1,2] => 4
[1,1,2,2,1] => 8
[1,1,2,3] => 4
[1,1,3,1,1] => 6
[1,1,3,2] => 6
[1,1,4,1] => 4
[1,1,5] => 1
[1,2,1,1,1,1] => 5
[1,2,1,1,2] => 5
[1,2,1,2,1] => 10
[1,2,1,3] => 5
[1,2,2,1,1] => 15
[1,2,2,2] => 15
[1,2,3,1] => 15
[1,2,4] => 5
[1,3,1,1,1] => 10
[1,3,1,2] => 10
[1,3,2,1] => 20
[1,3,3] => 10
[1,4,1,1] => 10
[1,4,2] => 10
[1,5,1] => 5
[1,6] => 1
[2,1,1,1,1,1] => 6
[2,1,1,1,2] => 6
[2,1,1,2,1] => 12
[2,1,1,3] => 6
[2,1,2,1,1] => 18
[2,1,2,2] => 18
[2,1,3,1] => 18
[2,1,4] => 6
[2,2,1,1,1] => 24
[2,2,1,2] => 24
[2,2,2,1] => 48
[2,2,3] => 24
[2,3,1,1] => 36
[2,3,2] => 36
[2,4,1] => 24
[2,5] => 6
[3,1,1,1,1] => 15
[3,1,1,2] => 15
[3,1,2,1] => 30
[3,1,3] => 15
[3,2,1,1] => 45
[3,2,2] => 45
[3,3,1] => 45
[3,4] => 15
[4,1,1,1] => 20
[4,1,2] => 20
[4,2,1] => 40
[4,3] => 20
[5,1,1] => 15
[5,2] => 15
[6,1] => 6
[7] => 1
[1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,2] => 1
[1,1,1,1,1,2,1] => 2
[1,1,1,1,1,3] => 1
[1,1,1,1,2,1,1] => 3
[1,1,1,1,2,2] => 3
[1,1,1,1,3,1] => 3
[1,1,1,1,4] => 1
[1,1,1,2,1,1,1] => 4
[1,1,1,2,1,2] => 4
[1,1,1,2,2,1] => 8
[1,1,1,2,3] => 4
[1,1,1,3,1,1] => 6
[1,1,1,3,2] => 6
[1,1,1,4,1] => 4
[1,1,1,5] => 1
[1,1,2,1,1,1,1] => 5
[1,1,2,1,1,2] => 5
[1,1,2,1,2,1] => 10
[1,1,2,1,3] => 5
[1,1,2,2,1,1] => 15
[1,1,2,2,2] => 15
[1,1,2,3,1] => 15
[1,1,2,4] => 5
[1,1,3,1,1,1] => 10
[1,1,3,1,2] => 10
[1,1,3,2,1] => 20
[1,1,3,3] => 10
[1,1,4,1,1] => 10
[1,1,4,2] => 10
[1,1,5,1] => 5
[1,1,6] => 1
[1,2,1,1,1,1,1] => 6
[1,2,1,1,1,2] => 6
[1,2,1,1,2,1] => 12
[1,2,1,1,3] => 6
[1,2,1,2,1,1] => 18
[1,2,1,2,2] => 18
[1,2,1,3,1] => 18
[1,2,1,4] => 6
[1,2,2,1,1,1] => 24
[1,2,2,1,2] => 24
[1,2,2,2,1] => 48
[1,2,2,3] => 24
[1,2,3,1,1] => 36
[1,2,3,2] => 36
[1,2,4,1] => 24
[1,2,5] => 6
[1,3,1,1,1,1] => 15
[1,3,1,1,2] => 15
[1,3,1,2,1] => 30
[1,3,1,3] => 15
[1,3,2,1,1] => 45
[1,3,2,2] => 45
[1,3,3,1] => 45
[1,3,4] => 15
[1,4,1,1,1] => 20
[1,4,1,2] => 20
[1,4,2,1] => 40
[1,4,3] => 20
[1,5,1,1] => 15
[1,5,2] => 15
[1,6,1] => 6
[1,7] => 1
[2,1,1,1,1,1,1] => 7
[2,1,1,1,1,2] => 7
[2,1,1,1,2,1] => 14
[2,1,1,2,1,1] => 21
[2,1,1,2,2] => 21
[2,1,1,3,1] => 21
[2,1,2,1,1,1] => 28
[2,1,2,1,2] => 28
[2,1,2,2,1] => 56
[2,1,2,3] => 28
[2,1,3,1,1] => 42
[2,1,3,2] => 42
[2,1,4,1] => 28
[2,2,1,1,1,1] => 35
[2,2,1,1,2] => 35
[2,2,1,2,1] => 70
[2,2,1,3] => 35
[2,2,2,1,1] => 105
[2,2,2,2] => 105
[2,2,3,1] => 105
[2,2,4] => 35
[2,3,1,1,1] => 70
[2,3,1,2] => 70
[2,3,2,1] => 140
[2,3,3] => 70
[2,4,2] => 70
[2,5,1] => 35
[2,6] => 7
[3,1,1,1,1,1] => 21
[3,1,1,3] => 21
[3,1,2,1,1] => 63
[3,1,2,2] => 63
[3,1,3,1] => 63
[3,2,1,1,1] => 84
[3,2,1,2] => 84
[3,2,2,1] => 168
[3,2,3] => 84
[3,3,1,1] => 126
[3,3,2] => 126
[3,4,1] => 84
[3,5] => 21
[4,1,1,1,1] => 35
[4,1,2,1] => 70
[4,2,1,1] => 105
[4,2,2] => 105
[4,3,1] => 105
[4,4] => 35
[5,1,1,1] => 35
[5,2,1] => 70
[5,3] => 35
[6,1,1] => 21
[6,2] => 21
[7,1] => 7
[8] => 1
[1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,2] => 1
[1,1,1,1,1,1,2,1] => 2
[1,1,1,1,1,1,3] => 1
[1,1,1,1,1,2,1,1] => 3
[1,1,1,1,1,2,2] => 3
[1,1,1,1,1,3,1] => 3
[1,1,1,1,1,4] => 1
[1,1,1,1,2,1,1,1] => 4
[1,1,1,1,2,1,2] => 4
[1,1,1,1,2,3] => 4
[1,1,1,1,3,1,1] => 6
[1,1,1,1,4,1] => 4
[1,1,1,1,5] => 1
[1,1,1,2,1,1,1,1] => 5
[1,1,1,2,1,1,2] => 5
[1,1,1,2,2,2] => 15
[1,1,1,2,3,1] => 15
[1,1,1,2,4] => 5
[1,1,1,3,1,1,1] => 10
[1,1,1,3,3] => 10
[1,1,1,4,2] => 10
[1,1,1,6] => 1
[1,1,2,1,1,1,1,1] => 6
[1,1,2,1,1,1,2] => 6
[1,1,2,1,3,1] => 18
[1,1,2,1,4] => 6
[1,1,2,2,1,2] => 24
[1,1,2,2,3] => 24
[1,1,2,3,2] => 36
[1,1,2,5] => 6
[1,1,3,1,1,1,1] => 15
[1,1,3,1,1,2] => 15
[1,1,3,1,3] => 15
[1,1,3,2,2] => 45
[1,1,3,4] => 15
[1,1,4,1,2] => 20
[1,1,6,1] => 6
[1,1,7] => 1
[1,2,1,1,1,1,1,1] => 7
[1,2,1,1,1,1,2] => 7
[1,2,1,1,2,1,1] => 21
[1,2,1,1,2,2] => 21
[1,2,1,1,3,1] => 21
[1,2,1,1,4] => 7
[1,2,1,2,1,2] => 28
[1,2,1,2,2,1] => 56
[1,2,1,2,3] => 28
[1,2,1,3,2] => 42
[1,2,2,1,1,2] => 35
[1,2,2,1,2,1] => 70
[1,2,2,1,3] => 35
[1,2,2,2,2] => 105
[1,2,2,4] => 35
[1,2,3,1,2] => 70
[1,2,3,3] => 70
[1,2,4,2] => 70
[1,2,5,1] => 35
[1,2,6] => 7
[1,3,1,1,1,1,1] => 21
[1,3,1,1,1,2] => 21
[1,3,1,1,2,1] => 42
[1,3,1,1,3] => 21
[1,3,1,2,1,1] => 63
[1,3,1,2,2] => 63
[1,3,2,1,1,1] => 84
[1,3,2,1,2] => 84
[1,3,2,3] => 84
[1,3,3,1,1] => 126
[1,3,3,2] => 126
[1,3,4,1] => 84
[1,3,5] => 21
[1,4,1,1,1,1] => 35
[1,4,1,1,2] => 35
[1,4,1,3] => 35
[1,4,2,2] => 105
[1,4,3,1] => 105
[1,4,4] => 35
[1,5,1,2] => 35
[1,5,2,1] => 70
[1,5,3] => 35
[1,6,1,1] => 21
[1,6,2] => 21
[1,7,1] => 7
[1,8] => 1
[2,1,1,1,1,1,1,1] => 8
[2,1,1,2,1,1,1] => 32
[2,1,2,2,1,1] => 120
[2,1,5,1] => 40
[2,2,1,1,1,1,1] => 48
[2,2,1,2,1,1] => 144
[2,2,2,1,1,1] => 192
[2,2,2,2,1] => 384
[2,2,2,3] => 192
[2,2,3,2] => 288
[2,2,4,1] => 192
[2,2,5] => 48
[2,3,2,2] => 360
[2,3,3,1] => 360
[2,3,4] => 120
[2,4,3] => 160
[2,7] => 8
[3,1,1,1,1,1,1] => 28
[3,1,4,1] => 112
[3,2,1,1,1,1] => 140
[3,2,2,1,1] => 420
[3,2,2,2] => 420
[3,3,1,1,1] => 280
[3,3,2,1] => 560
[3,3,3] => 280
[3,4,2] => 280
[3,6] => 28
[4,1,1,1,1,1] => 56
[4,2,1,1,1] => 224
[4,2,2,1] => 448
[4,3,1,1] => 336
[4,3,2] => 336
[4,4,1] => 224
[4,5] => 56
[5,1,1,1,1] => 70
[5,2,1,1] => 210
[5,2,2] => 210
[5,3,1] => 210
[5,4] => 70
[6,1,1,1] => 56
[6,2,1] => 112
[6,3] => 56
[7,1,1] => 28
[7,2] => 28
[8,1] => 8
[9] => 1
[1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,2] => 1
[1,1,1,1,1,1,1,2,1] => 2
[1,1,1,1,1,1,1,3] => 1
[1,1,1,1,1,1,2,1,1] => 3
[1,1,1,1,1,1,2,2] => 3
[1,1,1,1,1,1,4] => 1
[1,1,1,1,1,2,1,1,1] => 4
[1,1,1,1,1,2,3] => 4
[1,1,1,1,1,5] => 1
[1,1,1,1,2,1,1,1,1] => 5
[1,1,1,1,2,1,1,2] => 5
[1,1,1,1,2,2,1,1] => 15
[1,1,1,1,2,2,2] => 15
[1,1,1,1,2,4] => 5
[1,1,1,1,3,3] => 10
[1,1,1,1,5,1] => 5
[1,1,1,1,6] => 1
[1,1,1,2,1,1,1,1,1] => 6
[1,1,1,2,1,1,1,2] => 6
[1,1,1,2,2,3] => 24
[1,1,1,2,5] => 6
[1,1,1,3,4] => 15
[1,1,1,7] => 1
[1,1,2,1,1,1,1,1,1] => 7
[1,1,2,1,1,1,1,2] => 7
[1,1,2,1,1,2,1,1] => 21
[1,1,2,1,2,3] => 28
[1,1,2,2,1,1,1,1] => 35
[1,1,2,2,2,2] => 105
[1,1,2,2,4] => 35
[1,1,2,3,3] => 70
[1,1,2,6] => 7
[1,1,3,1,1,1,2] => 21
[1,1,3,1,1,3] => 21
[1,1,3,2,1,2] => 84
[1,1,3,3,1,1] => 126
[1,1,3,5] => 21
[1,1,4,4] => 35
[1,1,7,1] => 7
[1,1,8] => 1
[1,2,1,1,1,1,1,1,1] => 8
[1,2,1,1,1,1,1,2] => 8
[1,2,1,1,4,1] => 32
[1,2,1,2,1,1,2] => 40
[1,2,2,1,1,1,2] => 48
[1,2,2,1,2,1,1] => 144
[1,2,2,2,2,1] => 384
[1,2,2,2,3] => 192
[1,2,2,3,2] => 288
[1,2,2,5] => 48
[1,2,3,1,1,2] => 120
[1,2,3,2,2] => 360
[1,2,3,3,1] => 360
[1,2,3,4] => 120
[1,2,4,3] => 160
[1,2,6,1] => 48
[1,2,7] => 8
[1,3,1,1,1,1,2] => 28
-----------------------------------------------------------------------------
Created: Mar 24, 2013 at 23:08 by Chris Berg
-----------------------------------------------------------------------------
Last Updated: Jun 07, 2022 at 20:40 by Martin Rubey