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Statistic identifier: St000939
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Collection: Integer partitions
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Description: The number of characters of the symmetric group whose value on the partition is positive.
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References: [1] Miller, A. R. Note on parity and the irreducible characters of the symmetric group [[arXiv:1708.03267]]
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Code:
def statistic(la):
s = SymmetricFunctions(ZZ).s()
p = SymmetricFunctions(ZZ).p()
P = Partitions(la.size())
r = P.cardinality()
res = [0]*r
for i, (mu, v) in enumerate(s(p(la))):
res[i] = v
return len([1 for e in res if e > 0])
-----------------------------------------------------------------------------
Statistic values:
[2] => 1
[1,1] => 2
[3] => 2
[2,1] => 1
[1,1,1] => 3
[4] => 2
[3,1] => 2
[2,2] => 3
[2,1,1] => 2
[1,1,1,1] => 5
[5] => 3
[4,1] => 2
[3,2] => 3
[3,1,1] => 4
[2,2,1] => 4
[2,1,1,1] => 3
[1,1,1,1,1] => 7
[6] => 3
[5,1] => 3
[4,2] => 4
[4,1,1] => 4
[3,3] => 6
[3,2,1] => 3
[3,1,1,1] => 6
[2,2,2] => 5
[2,2,1,1] => 8
[2,1,1,1,1] => 5
[1,1,1,1,1,1] => 11
[7] => 4
[6,1] => 3
[5,2] => 5
[5,1,1] => 6
[4,3] => 6
[4,2,1] => 4
[4,1,1,1] => 6
[3,3,1] => 5
[3,2,2] => 7
[3,2,1,1] => 6
[3,1,1,1,1] => 9
[2,2,2,1] => 5
[2,2,1,1,1] => 10
[2,1,1,1,1,1] => 7
[1,1,1,1,1,1,1] => 15
[8] => 4
[7,1] => 4
[6,2] => 6
[6,1,1] => 6
[5,3] => 8
[5,2,1] => 5
[5,1,1,1] => 9
[4,4] => 8
[4,3,1] => 6
[4,2,2] => 7
[4,2,1,1] => 5
[4,1,1,1,1] => 8
[3,3,2] => 9
[3,3,1,1] => 12
[3,2,2,1] => 9
[3,2,1,1,1] => 9
[3,1,1,1,1,1] => 14
[2,2,2,2] => 10
[2,2,2,1,1] => 8
[2,2,1,1,1,1] => 15
[2,1,1,1,1,1,1] => 10
[1,1,1,1,1,1,1,1] => 22
[9] => 5
[8,1] => 4
[7,2] => 7
[7,1,1] => 8
[6,3] => 9
[6,2,1] => 6
[6,1,1,1] => 9
[5,4] => 10
[5,3,1] => 8
[5,2,2] => 13
[5,2,1,1] => 10
[5,1,1,1,1] => 15
[4,4,1] => 9
[4,3,2] => 10
[4,3,1,1] => 9
[4,2,2,1] => 7
[4,2,1,1,1] => 10
[4,1,1,1,1,1] => 12
[3,3,3] => 13
[3,3,2,1] => 9
[3,3,1,1,1] => 13
[3,2,2,2] => 13
[3,2,2,1,1] => 14
[3,2,1,1,1,1] => 13
[3,1,1,1,1,1,1] => 17
[2,2,2,2,1] => 13
[2,2,2,1,1,1] => 13
[2,2,1,1,1,1,1] => 21
[2,1,1,1,1,1,1,1] => 14
[1,1,1,1,1,1,1,1,1] => 30
[10] => 5
[9,1] => 5
[8,2] => 8
[8,1,1] => 8
[7,3] => 11
[7,2,1] => 7
[7,1,1,1] => 12
[6,4] => 12
[6,3,1] => 9
[6,2,2] => 15
[6,2,1,1] => 12
[6,1,1,1,1] => 15
[5,5] => 12
[5,4,1] => 10
[5,3,2] => 12
[5,3,1,1] => 13
[5,2,2,1] => 16
[5,2,1,1,1] => 13
[5,1,1,1,1,1] => 19
[4,4,2] => 14
[4,4,1,1] => 16
[4,3,3] => 18
[4,3,2,1] => 12
[4,3,1,1,1] => 18
[4,2,2,2] => 14
[4,2,2,1,1] => 14
[4,2,1,1,1,1] => 16
[4,1,1,1,1,1,1] => 20
[3,3,3,1] => 12
[3,3,2,2] => 17
[3,3,2,1,1] => 18
[3,3,1,1,1,1] => 23
[3,2,2,2,1] => 15
[3,2,2,1,1,1] => 19
[3,2,1,1,1,1,1] => 19
[3,1,1,1,1,1,1,1] => 25
[2,2,2,2,2] => 18
[2,2,2,2,1,1] => 24
[2,2,2,1,1,1,1] => 16
[2,2,1,1,1,1,1,1] => 28
[2,1,1,1,1,1,1,1,1] => 20
[1,1,1,1,1,1,1,1,1,1] => 42
[11] => 6
[10,1] => 5
[9,2] => 9
[9,1,1] => 10
[8,3] => 12
[8,2,1] => 8
[8,1,1,1] => 12
[7,4] => 14
[7,3,1] => 11
[7,2,2] => 18
[7,2,1,1] => 14
[7,1,1,1,1] => 20
[6,5] => 15
[6,4,1] => 12
[6,3,2] => 18
[6,3,1,1] => 18
[6,2,2,1] => 15
[6,2,1,1,1] => 18
[6,1,1,1,1,1] => 21
[5,5,1] => 11
[5,4,2] => 16
[5,4,1,1] => 17
[5,3,3] => 25
[5,3,2,1] => 15
[5,3,1,1,1] => 25
[5,2,2,2] => 23
[5,2,2,1,1] => 24
[5,2,1,1,1,1] => 23
[5,1,1,1,1,1,1] => 30
[4,4,3] => 25
[4,4,2,1] => 14
[4,4,1,1,1] => 25
[4,3,3,1] => 18
[4,3,2,2] => 21
[4,3,2,1,1] => 14
[4,3,1,1,1,1] => 24
[4,2,2,2,1] => 20
[4,2,2,1,1,1] => 21
[4,2,1,1,1,1,1] => 22
[4,1,1,1,1,1,1,1] => 26
[3,3,3,2] => 22
[3,3,3,1,1] => 26
[3,3,2,2,1] => 26
[3,3,2,1,1,1] => 17
[3,3,1,1,1,1,1] => 37
[3,2,2,2,2] => 25
[3,2,2,2,1,1] => 24
[3,2,2,1,1,1,1] => 32
[3,2,1,1,1,1,1,1] => 23
[3,1,1,1,1,1,1,1,1] => 39
[2,2,2,2,2,1] => 18
[2,2,2,2,1,1,1] => 31
[2,2,2,1,1,1,1,1] => 26
[2,2,1,1,1,1,1,1,1] => 41
[2,1,1,1,1,1,1,1,1,1] => 27
[1,1,1,1,1,1,1,1,1,1,1] => 56
[12] => 6
[11,1] => 6
[10,2] => 10
[10,1,1] => 10
[9,3] => 14
[9,2,1] => 9
[9,1,1,1] => 15
[8,4] => 16
[8,3,1] => 12
[8,2,2] => 20
[8,2,1,1] => 16
[8,1,1,1,1] => 20
[7,5] => 18
[7,4,1] => 14
[7,3,2] => 21
[7,3,1,1] => 22
[7,2,2,1] => 19
[7,2,1,1,1] => 21
[7,1,1,1,1,1] => 28
[6,6] => 15
[6,5,1] => 15
[6,4,2] => 21
[6,4,1,1] => 21
[6,3,3] => 18
[6,3,2,1] => 15
[6,3,1,1,1] => 20
[6,2,2,2] => 26
[6,2,2,1,1] => 23
[6,2,1,1,1,1] => 26
[6,1,1,1,1,1,1] => 27
[5,5,2] => 20
[5,5,1,1] => 24
[5,4,3] => 24
[5,4,2,1] => 20
[5,4,1,1,1] => 26
[5,3,3,1] => 25
[5,3,2,2] => 31
[5,3,2,1,1] => 24
[5,3,1,1,1,1] => 35
[5,2,2,2,1] => 25
[5,2,2,1,1,1] => 32
[5,2,1,1,1,1,1] => 30
[5,1,1,1,1,1,1,1] => 41
[4,4,4] => 20
[4,4,3,1] => 24
[4,4,2,2] => 28
[4,4,2,1,1] => 20
[4,4,1,1,1,1] => 32
[4,3,3,2] => 31
[4,3,3,1,1] => 29
[4,3,2,2,1] => 19
[4,3,2,1,1,1] => 27
[4,3,1,1,1,1,1] => 31
[4,2,2,2,2] => 29
[4,2,2,2,1,1] => 33
[4,2,2,1,1,1,1] => 25
[4,2,1,1,1,1,1,1] => 37
[4,1,1,1,1,1,1,1,1] => 34
[3,3,3,3] => 31
[3,3,3,2,1] => 20
[3,3,3,1,1,1] => 30
[3,3,2,2,2] => 32
[3,3,2,2,1,1] => 35
[3,3,2,1,1,1,1] => 30
[3,3,1,1,1,1,1,1] => 52
[3,2,2,2,2,1] => 29
[3,2,2,2,1,1,1] => 30
[3,2,2,1,1,1,1,1] => 40
[3,2,1,1,1,1,1,1,1] => 31
[3,1,1,1,1,1,1,1,1,1] => 53
[2,2,2,2,2,2] => 31
[2,2,2,2,2,1,1] => 28
[2,2,2,2,1,1,1,1] => 42
[2,2,2,1,1,1,1,1,1] => 35
[2,2,1,1,1,1,1,1,1,1] => 57
[2,1,1,1,1,1,1,1,1,1,1] => 37
[1,1,1,1,1,1,1,1,1,1,1,1] => 77
-----------------------------------------------------------------------------
Created: Aug 12, 2017 at 11:38 by Martin Rubey
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Last Updated: Aug 12, 2017 at 11:38 by Martin Rubey