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* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
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-----------------------------------------------------------------------------
Statistic identifier: St000816
-----------------------------------------------------------------------------
Collection: Integer compositions
-----------------------------------------------------------------------------
Description: The number of standard composition tableaux of the composition.
See [1, Def. 4.2.6].
Apparently, the total number of tableaux of given size is the number of involutions.
-----------------------------------------------------------------------------
References: [1] Luoto, K., Mykytiuk, S., van Willigenburg, S. An introduction to quasisymmetric Schur functions [[MathSciNet:3097867]]
-----------------------------------------------------------------------------
Code:
def statistic(c):
F = QuasiSymmetricFunctions(ZZ).F()
QS = QuasiSymmetricFunctions(ZZ).QS()
return sum(coeff for _, coeff in F(QS(c)))
-----------------------------------------------------------------------------
Statistic values:
[1] => 1
[1,1] => 1
[2] => 1
[1,1,1] => 1
[1,2] => 1
[2,1] => 1
[3] => 1
[1,1,1,1] => 1
[1,1,2] => 1
[1,2,1] => 1
[1,3] => 2
[2,1,1] => 1
[2,2] => 2
[3,1] => 1
[4] => 1
[1,1,1,1,1] => 1
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,1,3] => 3
[1,2,1,1] => 1
[1,2,2] => 2
[1,3,1] => 2
[1,4] => 3
[2,1,1,1] => 1
[2,1,2] => 1
[2,2,1] => 2
[2,3] => 2
[3,1,1] => 1
[3,2] => 3
[4,1] => 1
[5] => 1
[1,1,1,1,1,1] => 1
[1,1,1,1,2] => 1
[1,1,1,2,1] => 1
[1,1,1,3] => 4
[1,1,2,1,1] => 1
[1,1,2,2] => 2
[1,1,3,1] => 3
[1,1,4] => 6
[1,2,1,1,1] => 1
[1,2,1,2] => 1
[1,2,2,1] => 2
[1,2,3] => 2
[1,3,1,1] => 2
[1,3,2] => 5
[1,4,1] => 3
[1,5] => 4
[2,1,1,1,1] => 1
[2,1,1,2] => 1
[2,1,2,1] => 1
[2,1,3] => 3
[2,2,1,1] => 2
[2,2,2] => 5
[2,3,1] => 2
[2,4] => 5
[3,1,1,1] => 1
[3,1,2] => 1
[3,2,1] => 3
[3,3] => 5
[4,1,1] => 1
[4,2] => 4
[5,1] => 1
[6] => 1
[1,1,1,1,1,1,1] => 1
[1,1,1,1,1,2] => 1
[1,1,1,1,2,1] => 1
[1,1,1,1,3] => 5
[1,1,1,2,1,1] => 1
[1,1,1,2,2] => 2
[1,1,1,3,1] => 4
[1,1,1,4] => 10
[1,1,2,1,1,1] => 1
[1,1,2,1,2] => 1
[1,1,2,2,1] => 2
[1,1,2,3] => 2
[1,1,3,1,1] => 3
[1,1,3,2] => 7
[1,1,4,1] => 6
[1,1,5] => 10
[1,2,1,1,1,1] => 1
[1,2,1,1,2] => 1
[1,2,1,2,1] => 1
[1,2,1,3] => 3
[1,2,2,1,1] => 2
[1,2,2,2] => 5
[1,2,3,1] => 2
[1,2,4] => 7
[1,3,1,1,1] => 2
[1,3,1,2] => 2
[1,3,2,1] => 5
[1,3,3] => 12
[1,4,1,1] => 3
[1,4,2] => 9
[1,5,1] => 4
[1,6] => 5
[2,1,1,1,1,1] => 1
[2,1,1,1,2] => 1
[2,1,1,2,1] => 1
[2,1,1,3] => 4
[2,1,2,1,1] => 1
[2,1,2,2] => 2
[2,1,3,1] => 3
[2,1,4] => 9
[2,2,1,1,1] => 2
[2,2,1,2] => 2
[2,2,2,1] => 5
[2,2,3] => 5
[2,3,1,1] => 2
[2,3,2] => 7
[2,4,1] => 5
[2,5] => 9
[3,1,1,1,1] => 1
[3,1,1,2] => 1
[3,1,2,1] => 1
[3,1,3] => 4
[3,2,1,1] => 3
[3,2,2] => 9
[3,3,1] => 5
[3,4] => 5
[4,1,1,1] => 1
[4,1,2] => 1
[4,2,1] => 4
[4,3] => 9
[5,1,1] => 1
[5,2] => 5
[6,1] => 1
[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] => 1
[1,1,1,1,1,3] => 6
[1,1,1,1,2,1,1] => 1
[1,1,1,1,2,2] => 2
[1,1,1,1,3,1] => 5
[1,1,1,1,4] => 15
[1,1,1,2,1,1,1] => 1
[1,1,1,2,1,2] => 1
[1,1,1,2,2,1] => 2
[1,1,1,2,3] => 2
[1,1,1,3,1,1] => 4
[1,1,1,3,2] => 9
[1,1,1,4,1] => 10
[1,1,1,5] => 20
[1,1,2,1,1,1,1] => 1
[1,1,2,1,1,2] => 1
[1,1,2,1,2,1] => 1
[1,1,2,1,3] => 3
[1,1,2,2,1,1] => 2
[1,1,2,2,2] => 5
[1,1,2,3,1] => 2
[1,1,2,4] => 9
[1,1,3,1,1,1] => 3
[1,1,3,1,2] => 3
[1,1,3,2,1] => 7
[1,1,3,3] => 21
[1,1,4,1,1] => 6
[1,1,4,2] => 16
[1,1,5,1] => 10
[1,1,6] => 15
[1,2,1,1,1,1,1] => 1
[1,2,1,1,1,2] => 1
[1,2,1,1,2,1] => 1
[1,2,1,1,3] => 4
[1,2,1,2,1,1] => 1
[1,2,1,2,2] => 2
[1,2,1,3,1] => 3
[1,2,1,4] => 12
[1,2,2,1,1,1] => 2
[1,2,2,1,2] => 2
[1,2,2,2,1] => 5
[1,2,2,3] => 5
[1,2,3,1,1] => 2
[1,2,3,2] => 7
[1,2,4,1] => 7
[1,2,5] => 16
[1,3,1,1,1,1] => 2
[1,3,1,1,2] => 2
[1,3,1,2,1] => 2
[1,3,1,3] => 9
[1,3,2,1,1] => 5
[1,3,2,2] => 14
[1,3,3,1] => 12
[1,3,4] => 12
[1,4,1,1,1] => 3
[1,4,1,2] => 3
[1,4,2,1] => 9
[1,4,3] => 30
[1,5,1,1] => 4
[1,5,2] => 14
[1,6,1] => 5
[1,7] => 6
[2,1,1,1,1,1,1] => 1
[2,1,1,1,1,2] => 1
[2,1,1,1,2,1] => 1
[2,1,1,1,3] => 5
[2,1,1,2,1,1] => 1
[2,1,1,2,2] => 2
[2,1,1,3,1] => 4
[2,1,1,4] => 14
[2,1,2,1,1,1] => 1
[2,1,2,1,2] => 1
[2,1,2,2,1] => 2
[2,1,2,3] => 2
[2,1,3,1,1] => 3
[2,1,3,2] => 7
[2,1,4,1] => 9
[2,1,5] => 19
[2,2,1,1,1,1] => 2
[2,2,1,1,2] => 2
[2,2,1,2,1] => 2
[2,2,1,3] => 7
[2,2,2,1,1] => 5
[2,2,2,2] => 14
[2,2,3,1] => 5
[2,2,4] => 21
[2,3,1,1,1] => 2
[2,3,1,2] => 2
[2,3,2,1] => 7
[2,3,3] => 12
[2,4,1,1] => 5
[2,4,2] => 21
[2,5,1] => 9
[2,6] => 14
[3,1,1,1,1,1] => 1
[3,1,1,1,2] => 1
[3,1,1,2,1] => 1
[3,1,1,3] => 5
[3,1,2,1,1] => 1
[3,1,2,2] => 2
[3,1,3,1] => 4
[3,1,4] => 9
[3,2,1,1,1] => 3
[3,2,1,2] => 3
[3,2,2,1] => 9
[3,2,3] => 9
[3,3,1,1] => 5
[3,3,2] => 21
[3,4,1] => 5
[3,5] => 14
[4,1,1,1,1] => 1
[4,1,1,2] => 1
[4,1,2,1] => 1
[4,1,3] => 5
[4,2,1,1] => 4
[4,2,2] => 14
[4,3,1] => 9
[4,4] => 14
[5,1,1,1] => 1
[5,1,2] => 1
[5,2,1] => 5
[5,3] => 14
[6,1,1] => 1
[6,2] => 6
[7,1] => 1
[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] => 1
[1,1,1,1,1,1,3] => 7
[1,1,1,1,1,2,1,1] => 1
[1,1,1,1,1,2,2] => 2
[1,1,1,1,1,3,1] => 6
[1,1,1,1,1,4] => 21
[1,1,1,1,2,1,1,1] => 1
[1,1,1,1,2,1,2] => 1
[1,1,1,1,2,2,1] => 2
[1,1,1,1,2,3] => 2
[1,1,1,1,3,1,1] => 5
[1,1,1,1,3,2] => 11
[1,1,1,1,4,1] => 15
[1,1,1,1,5] => 35
[1,1,1,2,1,1,1,1] => 1
[1,1,1,2,1,1,2] => 1
[1,1,1,2,1,2,1] => 1
[1,1,1,2,1,3] => 3
[1,1,1,2,2,1,1] => 2
[1,1,1,2,2,2] => 5
[1,1,1,2,3,1] => 2
[1,1,1,2,4] => 11
[1,1,1,3,1,1,1] => 4
[1,1,1,3,1,2] => 4
[1,1,1,3,2,1] => 9
[1,1,1,3,3] => 32
[1,1,1,4,1,1] => 10
[1,1,1,4,2] => 25
[1,1,1,5,1] => 20
[1,1,1,6] => 35
[1,1,2,1,1,1,1,1] => 1
[1,1,2,1,1,1,2] => 1
[1,1,2,1,1,2,1] => 1
[1,1,2,1,1,3] => 4
[1,1,2,1,2,1,1] => 1
[1,1,2,1,2,2] => 2
[1,1,2,1,3,1] => 3
[1,1,2,1,4] => 15
[1,1,2,2,1,1,1] => 2
[1,1,2,2,1,2] => 2
[1,1,2,2,2,1] => 5
[1,1,2,2,3] => 5
[1,1,2,3,1,1] => 2
[1,1,2,3,2] => 7
[1,1,2,4,1] => 9
[1,1,2,5] => 25
[1,1,3,1,1,1,1] => 3
[1,1,3,1,1,2] => 3
[1,1,3,1,2,1] => 3
[1,1,3,1,3] => 15
[1,1,3,2,1,1] => 7
[1,1,3,2,2] => 19
[1,1,3,3,1] => 21
[1,1,3,4] => 21
[1,1,4,1,1,1] => 6
[1,1,4,1,2] => 6
[1,1,4,2,1] => 16
[1,1,4,3] => 67
[1,1,5,1,1] => 10
[1,1,5,2] => 30
[1,1,6,1] => 15
[1,1,7] => 21
[1,2,1,1,1,1,1,1] => 1
[1,2,1,1,1,1,2] => 1
[1,2,1,1,1,2,1] => 1
[1,2,1,1,1,3] => 5
[1,2,1,1,2,1,1] => 1
[1,2,1,1,2,2] => 2
[1,2,1,1,3,1] => 4
[1,2,1,1,4] => 18
[1,2,1,2,1,1,1] => 1
[1,2,1,2,1,2] => 1
[1,2,1,2,2,1] => 2
[1,2,1,2,3] => 2
[1,2,1,3,1,1] => 3
[1,2,1,3,2] => 7
[1,2,1,4,1] => 12
[1,2,1,5] => 31
[1,2,2,1,1,1,1] => 2
[1,2,2,1,1,2] => 2
[1,2,2,1,2,1] => 2
[1,2,2,1,3] => 7
[1,2,2,2,1,1] => 5
[1,2,2,2,2] => 14
[1,2,2,3,1] => 5
[1,2,2,4] => 26
[1,2,3,1,1,1] => 2
[1,2,3,1,2] => 2
[1,2,3,2,1] => 7
[1,2,3,3] => 12
[1,2,4,1,1] => 7
[1,2,4,2] => 28
[1,2,5,1] => 16
[1,2,6] => 30
[1,3,1,1,1,1,1] => 2
[1,3,1,1,1,2] => 2
[1,3,1,1,2,1] => 2
[1,3,1,1,3] => 11
[1,3,1,2,1,1] => 2
[1,3,1,2,2] => 4
[1,3,1,3,1] => 9
[1,3,1,4] => 21
[1,3,2,1,1,1] => 5
[1,3,2,1,2] => 5
[1,3,2,2,1] => 14
[1,3,2,3] => 14
[1,3,3,1,1] => 12
[1,3,3,2] => 42
[1,3,4,1] => 12
[1,3,5] => 42
[1,4,1,1,1,1] => 3
[1,4,1,1,2] => 3
[1,4,1,2,1] => 3
[1,4,1,3] => 17
[1,4,2,1,1] => 9
[1,4,2,2] => 28
[1,4,3,1] => 30
[1,4,4] => 56
[1,5,1,1,1] => 4
[1,5,1,2] => 4
[1,5,2,1] => 14
[1,5,3] => 58
[1,6,1,1] => 5
[1,6,2] => 20
[1,7,1] => 6
[1,8] => 7
[2,1,1,1,1,1,1,1] => 1
[2,1,1,1,1,1,2] => 1
[2,1,1,1,1,2,1] => 1
[2,1,1,1,1,3] => 6
[2,1,1,1,2,1,1] => 1
[2,1,1,1,2,2] => 2
[2,1,1,1,3,1] => 5
[2,1,1,1,4] => 20
[2,1,1,2,1,1,1] => 1
[2,1,1,2,1,2] => 1
[2,1,1,2,2,1] => 2
[2,1,1,2,3] => 2
[2,1,1,3,1,1] => 4
[2,1,1,3,2] => 9
[2,1,1,4,1] => 14
[2,1,1,5] => 34
[2,1,2,1,1,1,1] => 1
[2,1,2,1,1,2] => 1
[2,1,2,1,2,1] => 1
[2,1,2,1,3] => 3
[2,1,2,2,1,1] => 2
[2,1,2,2,2] => 5
[2,1,2,3,1] => 2
[2,1,2,4] => 11
[2,1,3,1,1,1] => 3
[2,1,3,1,2] => 3
[2,1,3,2,1] => 7
[2,1,3,3] => 21
[2,1,4,1,1] => 9
[2,1,4,2] => 23
[2,1,5,1] => 19
[2,1,6] => 34
[2,2,1,1,1,1,1] => 2
[2,2,1,1,1,2] => 2
[2,2,1,1,2,1] => 2
[2,2,1,1,3] => 9
[2,2,1,2,1,1] => 2
[2,2,1,2,2] => 4
[2,2,1,3,1] => 7
[2,2,1,4] => 33
[2,2,2,1,1,1] => 5
[2,2,2,1,2] => 5
[2,2,2,2,1] => 14
[2,2,2,3] => 14
[2,2,3,1,1] => 5
[2,2,3,2] => 19
[2,2,4,1] => 21
[2,2,5] => 56
[2,3,1,1,1,1] => 2
[2,3,1,1,2] => 2
[2,3,1,2,1] => 2
[2,3,1,3] => 9
[2,3,2,1,1] => 7
[2,3,2,2] => 23
[2,3,3,1] => 12
[2,3,4] => 12
[2,4,1,1,1] => 5
[2,4,1,2] => 5
[2,4,2,1] => 21
[2,4,3] => 42
[2,5,1,1] => 9
[2,5,2] => 44
[2,6,1] => 14
[2,7] => 20
[3,1,1,1,1,1,1] => 1
[3,1,1,1,1,2] => 1
[3,1,1,1,2,1] => 1
[3,1,1,1,3] => 6
[3,1,1,2,1,1] => 1
[3,1,1,2,2] => 2
[3,1,1,3,1] => 5
[3,1,1,4] => 14
[3,1,2,1,1,1] => 1
[3,1,2,1,2] => 1
[3,1,2,2,1] => 2
[3,1,2,3] => 2
[3,1,3,1,1] => 4
[3,1,3,2] => 9
[3,1,4,1] => 9
[3,1,5] => 28
[3,2,1,1,1,1] => 3
[3,2,1,1,2] => 3
[3,2,1,2,1] => 3
[3,2,1,3] => 12
[3,2,2,1,1] => 9
[3,2,2,2] => 28
[3,2,3,1] => 9
[3,2,4] => 30
[3,3,1,1,1] => 5
[3,3,1,2] => 5
[3,3,2,1] => 21
[3,3,3] => 42
[3,4,1,1] => 5
[3,4,2] => 26
[3,5,1] => 14
[3,6] => 28
[4,1,1,1,1,1] => 1
[4,1,1,1,2] => 1
[4,1,1,2,1] => 1
[4,1,1,3] => 6
[4,1,2,1,1] => 1
[4,1,2,2] => 2
[4,1,3,1] => 5
[4,1,4] => 14
[4,2,1,1,1] => 4
[4,2,1,2] => 4
[4,2,2,1] => 14
[4,2,3] => 14
[4,3,1,1] => 9
[4,3,2] => 44
[4,4,1] => 14
[4,5] => 14
[5,1,1,1,1] => 1
[5,1,1,2] => 1
[5,1,2,1] => 1
[5,1,3] => 6
[5,2,1,1] => 5
[5,2,2] => 20
[5,3,1] => 14
[5,4] => 28
[6,1,1,1] => 1
[6,1,2] => 1
[6,2,1] => 6
[6,3] => 20
[7,1,1] => 1
[7,2] => 7
[8,1] => 1
[9] => 1
-----------------------------------------------------------------------------
Created: May 19, 2017 at 23:08 by Martin Rubey
-----------------------------------------------------------------------------
Last Updated: Nov 19, 2017 at 22:03 by Christian Stump