Identifier
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
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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.
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)))
Created
May 19, 2017 at 23:08 by Martin Rubey
Updated
Nov 19, 2017 at 22:03 by Christian Stump
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