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Statistic identifier: St001629
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Collection: Integer compositions
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Description: The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
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References:
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Code:
def statistic(a):
F = QuasiSymmetricFunctions(QQ).F()
S = species.CycleSpecies().cycle_index_series()
return F(S.coefficient(a.size())).coefficient(a)
-----------------------------------------------------------------------------
Statistic values:
[1,1,1] => 1
[1,2] => 0
[2,1] => 0
[3] => 1
[1,1,1,1] => 0
[1,1,2] => 1
[1,2,1] => 2
[1,3] => 0
[2,1,1] => 1
[2,2] => 1
[3,1] => 0
[4] => 1
[1,1,1,1,1] => 1
[1,1,1,2] => 0
[1,1,2,1] => 1
[1,1,3] => 2
[1,2,1,1] => 1
[1,2,2] => 4
[1,3,1] => 3
[1,4] => 0
[2,1,1,1] => 0
[2,1,2] => 3
[2,2,1] => 4
[2,3] => 1
[3,1,1] => 2
[3,2] => 1
[4,1] => 0
[5] => 1
[1,1,1,1,1,1] => 0
[1,1,1,1,2] => 1
[1,1,1,2,1] => 2
[1,1,1,3] => 2
[1,1,2,1,1] => 4
[1,1,2,2] => 5
[1,1,3,1] => 4
[1,1,4] => 2
[1,2,1,1,1] => 2
[1,2,1,2] => 7
[1,2,2,1] => 10
[1,2,3] => 6
[1,3,1,1] => 4
[1,3,2] => 7
[1,4,1] => 4
[1,5] => 0
[2,1,1,1,1] => 1
[2,1,1,2] => 3
[2,1,2,1] => 7
[2,1,3] => 4
[2,2,1,1] => 5
[2,2,2] => 11
[2,3,1] => 7
[2,4] => 2
[3,1,1,1] => 2
[3,1,2] => 4
[3,2,1] => 6
[3,3] => 3
[4,1,1] => 2
[4,2] => 2
[5,1] => 0
[6] => 1
[1,1,1,1,1,1,1] => 1
[1,1,1,1,1,2] => 0
[1,1,1,1,2,1] => 2
[1,1,1,1,3] => 3
[1,1,1,2,1,1] => 4
[1,1,1,2,2] => 10
[1,1,1,3,1] => 8
[1,1,1,4] => 2
[1,1,2,1,1,1] => 4
[1,1,2,1,2] => 15
[1,1,2,2,1] => 23
[1,1,2,3] => 12
[1,1,3,1,1] => 11
[1,1,3,2] => 15
[1,1,4,1] => 7
[1,1,5] => 3
[1,2,1,1,1,1] => 2
[1,2,1,1,2] => 12
[1,2,1,2,1] => 25
[1,2,1,3] => 15
[1,2,2,1,1] => 23
[1,2,2,2] => 38
[1,2,3,1] => 25
[1,2,4] => 10
[1,3,1,1,1] => 8
[1,3,1,2] => 18
[1,3,2,1] => 25
[1,3,3] => 15
[1,4,1,1] => 7
[1,4,2] => 12
[1,5,1] => 5
[1,6] => 0
[2,1,1,1,1,1] => 0
[2,1,1,1,2] => 5
[2,1,1,2,1] => 12
[2,1,1,3] => 7
[2,1,2,1,1] => 15
[2,1,2,2] => 25
[2,1,3,1] => 18
[2,1,4] => 8
[2,2,1,1,1] => 10
[2,2,1,2] => 25
[2,2,2,1] => 38
[2,2,3] => 23
[2,3,1,1] => 15
[2,3,2] => 25
[2,4,1] => 12
[2,5] => 2
[3,1,1,1,1] => 3
[3,1,1,2] => 7
[3,1,2,1] => 15
[3,1,3] => 11
[3,2,1,1] => 12
[3,2,2] => 23
[3,3,1] => 15
[3,4] => 4
[4,1,1,1] => 2
[4,1,2] => 8
[4,2,1] => 10
[4,3] => 4
[5,1,1] => 3
[5,2] => 2
[6,1] => 0
[7] => 1
[1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,2] => 1
[1,1,1,1,1,2,1] => 4
[1,1,1,1,1,3] => 2
[1,1,1,1,2,1,1] => 7
[1,1,1,1,2,2] => 13
[1,1,1,1,3,1] => 10
[1,1,1,1,4] => 5
[1,1,1,2,1,1,1] => 10
[1,1,1,2,1,2] => 24
[1,1,1,2,2,1] => 40
[1,1,1,2,3] => 24
[1,1,1,3,1,1] => 18
[1,1,1,3,2] => 32
[1,1,1,4,1] => 16
[1,1,1,5] => 4
[1,1,2,1,1,1,1] => 7
[1,1,2,1,1,2] => 27
[1,1,2,1,2,1] => 59
[1,1,2,1,3] => 40
[1,1,2,2,1,1] => 56
[1,1,2,2,2] => 99
[1,1,2,3,1] => 67
[1,1,2,4] => 23
[1,1,3,1,1,1] => 18
[1,1,3,1,2] => 53
[1,1,3,2,1] => 72
[1,1,3,3] => 39
[1,1,4,1,1] => 24
[1,1,4,2] => 31
[1,1,5,1] => 12
[1,1,6] => 3
[1,2,1,1,1,1,1] => 4
[1,2,1,1,1,2] => 16
[1,2,1,1,2,1] => 46
[1,2,1,1,3] => 32
[1,2,1,2,1,1] => 59
[1,2,1,2,2] => 110
[1,2,1,3,1] => 80
[1,2,1,4] => 31
[1,2,2,1,1,1] => 40
[1,2,2,1,2] => 115
[1,2,2,2,1] => 174
[1,2,2,3] => 98
[1,2,3,1,1] => 72
[1,2,3,2] => 109
[1,2,4,1] => 50
[1,2,5] => 14
[1,3,1,1,1,1] => 10
[1,3,1,1,2] => 40
[1,3,1,2,1] => 80
[1,3,1,3] => 52
[1,3,2,1,1] => 67
[1,3,2,2] => 114
[1,3,3,1] => 74
[1,3,4] => 25
[1,4,1,1,1] => 16
[1,4,1,2] => 39
[1,4,2,1] => 50
[1,4,3] => 28
[1,5,1,1] => 12
[1,5,2] => 17
[1,6,1] => 6
[1,7] => 0
[2,1,1,1,1,1,1] => 1
[2,1,1,1,1,2] => 5
[2,1,1,1,2,1] => 16
[2,1,1,1,3] => 13
[2,1,1,2,1,1] => 27
[2,1,1,2,2] => 51
[2,1,1,3,1] => 40
[2,1,1,4] => 15
[2,1,2,1,1,1] => 24
[2,1,2,1,2] => 75
[2,1,2,2,1] => 115
[2,1,2,3] => 66
[2,1,3,1,1] => 53
[2,1,3,2] => 79
[2,1,4,1] => 39
[2,1,5] => 11
[2,2,1,1,1,1] => 13
[2,2,1,1,2] => 51
[2,2,1,2,1] => 110
[2,2,1,3] => 71
[2,2,2,1,1] => 99
[2,2,2,2] => 173
[2,2,3,1] => 114
[2,2,4] => 41
[2,3,1,1,1] => 32
[2,3,1,2] => 79
[2,3,2,1] => 109
[2,3,3] => 60
[2,4,1,1] => 31
[2,4,2] => 47
[2,5,1] => 17
[2,6] => 3
[3,1,1,1,1,1] => 2
[3,1,1,1,2] => 13
[3,1,1,2,1] => 32
[3,1,1,3] => 23
[3,1,2,1,1] => 40
[3,1,2,2] => 71
[3,1,3,1] => 52
[3,1,4] => 19
[3,2,1,1,1] => 24
[3,2,1,2] => 66
[3,2,2,1] => 98
[3,2,3] => 57
[3,3,1,1] => 39
[3,3,2] => 60
[3,4,1] => 28
[3,5] => 6
[4,1,1,1,1] => 5
[4,1,1,2] => 15
[4,1,2,1] => 31
[4,1,3] => 19
[4,2,1,1] => 23
[4,2,2] => 41
[4,3,1] => 25
[4,4] => 9
[5,1,1,1] => 4
[5,1,2] => 11
[5,2,1] => 14
[5,3] => 6
[6,1,1] => 3
[6,2] => 3
[7,1] => 0
[8] => 1
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Created: Oct 02, 2020 at 07:18 by Martin Rubey
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Last Updated: Oct 02, 2020 at 07:18 by Martin Rubey