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Statistic identifier: St000938
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Collection: Integer partitions
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Description: The number of zeros of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
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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:
@cached_function
def table(n):
s = SymmetricFunctions(ZZ).s()
p = SymmetricFunctions(ZZ).p()
res = dict()
P = Partitions(n)
r = P.cardinality()
for mu in P:
res[mu] = [0]*r
for i, la in enumerate(P):
for mu, v in s(p(la)):
res[mu][i] = v
return res
def statistic(la):
t = table(la.size())
return len([1 for e in t[la] if e == 0])
-----------------------------------------------------------------------------
Statistic values:
[2] => 0
[1,1] => 0
[3] => 0
[2,1] => 1
[1,1,1] => 0
[4] => 0
[3,1] => 1
[2,2] => 2
[2,1,1] => 1
[1,1,1,1] => 0
[5] => 0
[4,1] => 2
[3,2] => 1
[3,1,1] => 4
[2,2,1] => 1
[2,1,1,1] => 2
[1,1,1,1,1] => 0
[6] => 0
[5,1] => 2
[4,2] => 4
[4,1,1] => 3
[3,3] => 2
[3,2,1] => 7
[3,1,1,1] => 3
[2,2,2] => 2
[2,2,1,1] => 4
[2,1,1,1,1] => 2
[1,1,1,1,1,1] => 0
[7] => 0
[6,1] => 4
[5,2] => 5
[5,1,1] => 4
[4,3] => 4
[4,2,1] => 3
[4,1,1,1] => 9
[3,3,1] => 3
[3,2,2] => 3
[3,2,1,1] => 3
[3,1,1,1,1] => 4
[2,2,2,1] => 4
[2,2,1,1,1] => 5
[2,1,1,1,1,1] => 4
[1,1,1,1,1,1,1] => 0
[8] => 0
[7,1] => 4
[6,2] => 6
[6,1,1] => 7
[5,3] => 5
[5,2,1] => 12
[5,1,1,1] => 6
[4,4] => 6
[4,3,1] => 7
[4,2,2] => 9
[4,2,1,1] => 16
[4,1,1,1,1] => 6
[3,3,2] => 13
[3,3,1,1] => 9
[3,2,2,1] => 7
[3,2,1,1,1] => 12
[3,1,1,1,1,1] => 7
[2,2,2,2] => 6
[2,2,2,1,1] => 5
[2,2,1,1,1,1] => 6
[2,1,1,1,1,1,1] => 4
[1,1,1,1,1,1,1,1] => 0
[9] => 0
[8,1] => 7
[7,2] => 9
[7,1,1] => 8
[6,3] => 13
[6,2,1] => 8
[6,1,1,1] => 9
[5,4] => 10
[5,3,1] => 17
[5,2,2] => 16
[5,2,1,1] => 9
[5,1,1,1,1] => 19
[4,4,1] => 8
[4,3,2] => 10
[4,3,1,1] => 11
[4,2,2,1] => 11
[4,2,1,1,1] => 9
[4,1,1,1,1,1] => 9
[3,3,3] => 18
[3,3,2,1] => 10
[3,3,1,1,1] => 16
[3,2,2,2] => 8
[3,2,2,1,1] => 17
[3,2,1,1,1,1] => 8
[3,1,1,1,1,1,1] => 8
[2,2,2,2,1] => 10
[2,2,2,1,1,1] => 13
[2,2,1,1,1,1,1] => 9
[2,1,1,1,1,1,1,1] => 7
[1,1,1,1,1,1,1,1,1] => 0
[10] => 0
[9,1] => 8
[8,2] => 11
[8,1,1] => 12
[7,3] => 14
[7,2,1] => 19
[7,1,1,1] => 11
[6,4] => 11
[6,3,1] => 8
[6,2,2] => 15
[6,2,1,1] => 13
[6,1,1,1,1] => 14
[5,5] => 13
[5,4,1] => 20
[5,3,2] => 14
[5,3,1,1] => 23
[5,2,2,1] => 15
[5,2,1,1,1] => 30
[5,1,1,1,1,1] => 14
[4,4,2] => 13
[4,4,1,1] => 14
[4,3,3] => 14
[4,3,2,1] => 34
[4,3,1,1,1] => 15
[4,2,2,2] => 14
[4,2,2,1,1] => 23
[4,2,1,1,1,1] => 13
[4,1,1,1,1,1,1] => 11
[3,3,3,1] => 14
[3,3,2,2] => 13
[3,3,2,1,1] => 14
[3,3,1,1,1,1] => 15
[3,2,2,2,1] => 20
[3,2,2,1,1,1] => 8
[3,2,1,1,1,1,1] => 19
[3,1,1,1,1,1,1,1] => 12
[2,2,2,2,2] => 13
[2,2,2,2,1,1] => 11
[2,2,2,1,1,1,1] => 14
[2,2,1,1,1,1,1,1] => 11
[2,1,1,1,1,1,1,1,1] => 8
[1,1,1,1,1,1,1,1,1,1] => 0
[11] => 0
[10,1] => 12
[9,2] => 14
[9,1,1] => 15
[8,3] => 16
[8,2,1] => 15
[8,1,1,1] => 19
[7,4] => 7
[7,3,1] => 21
[7,2,2] => 17
[7,2,1,1] => 28
[7,1,1,1,1] => 15
[6,5] => 19
[6,4,1] => 17
[6,3,2] => 21
[6,3,1,1] => 30
[6,2,2,1] => 29
[6,2,1,1,1] => 15
[6,1,1,1,1,1] => 37
[5,5,1] => 14
[5,4,2] => 24
[5,4,1,1] => 12
[5,3,3] => 15
[5,3,2,1] => 18
[5,3,1,1,1] => 16
[5,2,2,2] => 25
[5,2,2,1,1] => 16
[5,2,1,1,1,1] => 15
[5,1,1,1,1,1,1] => 15
[4,4,3] => 16
[4,4,2,1] => 21
[4,4,1,1,1] => 25
[4,3,3,1] => 39
[4,3,2,2] => 21
[4,3,2,1,1] => 18
[4,3,1,1,1,1] => 29
[4,2,2,2,1] => 12
[4,2,2,1,1,1] => 30
[4,2,1,1,1,1,1] => 28
[4,1,1,1,1,1,1,1] => 19
[3,3,3,2] => 16
[3,3,3,1,1] => 15
[3,3,2,2,1] => 24
[3,3,2,1,1,1] => 21
[3,3,1,1,1,1,1] => 17
[3,2,2,2,2] => 14
[3,2,2,2,1,1] => 17
[3,2,2,1,1,1,1] => 21
[3,2,1,1,1,1,1,1] => 15
[3,1,1,1,1,1,1,1,1] => 15
[2,2,2,2,2,1] => 19
[2,2,2,2,1,1,1] => 7
[2,2,2,1,1,1,1,1] => 16
[2,2,1,1,1,1,1,1,1] => 14
[2,1,1,1,1,1,1,1,1,1] => 12
[1,1,1,1,1,1,1,1,1,1,1] => 0
[12] => 0
[11,1] => 14
[10,2] => 20
[10,1,1] => 18
[9,3] => 18
[9,2,1] => 31
[9,1,1,1] => 21
[8,4] => 23
[8,3,1] => 30
[8,2,2] => 24
[8,2,1,1] => 22
[8,1,1,1,1] => 19
[7,5] => 18
[7,4,1] => 42
[7,3,2] => 31
[7,3,1,1] => 27
[7,2,2,1] => 28
[7,2,1,1,1] => 38
[7,1,1,1,1,1] => 23
[6,6] => 26
[6,5,1] => 18
[6,4,2] => 46
[6,4,1,1] => 30
[6,3,3] => 27
[6,3,2,1] => 55
[6,3,1,1,1] => 41
[6,2,2,2] => 35
[6,2,2,1,1] => 41
[6,2,1,1,1,1] => 53
[6,1,1,1,1,1,1] => 23
[5,5,2] => 26
[5,5,1,1] => 29
[5,4,3] => 34
[5,4,2,1] => 34
[5,4,1,1,1] => 36
[5,3,3,1] => 26
[5,3,2,2] => 35
[5,3,2,1,1] => 53
[5,3,1,1,1,1] => 41
[5,2,2,2,1] => 36
[5,2,2,1,1,1] => 41
[5,2,1,1,1,1,1] => 38
[5,1,1,1,1,1,1,1] => 19
[4,4,4] => 24
[4,4,3,1] => 25
[4,4,2,2] => 54
[4,4,2,1,1] => 35
[4,4,1,1,1,1] => 35
[4,3,3,2] => 25
[4,3,3,1,1] => 26
[4,3,2,2,1] => 34
[4,3,2,1,1,1] => 55
[4,3,1,1,1,1,1] => 28
[4,2,2,2,2] => 29
[4,2,2,2,1,1] => 30
[4,2,2,1,1,1,1] => 27
[4,2,1,1,1,1,1,1] => 22
[4,1,1,1,1,1,1,1,1] => 21
[3,3,3,3] => 24
[3,3,3,2,1] => 34
[3,3,3,1,1,1] => 27
[3,3,2,2,2] => 26
[3,3,2,2,1,1] => 46
[3,3,2,1,1,1,1] => 31
[3,3,1,1,1,1,1,1] => 24
[3,2,2,2,2,1] => 18
[3,2,2,2,1,1,1] => 42
[3,2,2,1,1,1,1,1] => 30
[3,2,1,1,1,1,1,1,1] => 31
[3,1,1,1,1,1,1,1,1,1] => 18
[2,2,2,2,2,2] => 26
[2,2,2,2,2,1,1] => 18
[2,2,2,2,1,1,1,1] => 23
[2,2,2,1,1,1,1,1,1] => 18
[2,2,1,1,1,1,1,1,1,1] => 20
[2,1,1,1,1,1,1,1,1,1,1] => 14
[1,1,1,1,1,1,1,1,1,1,1,1] => 0
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Created: Aug 11, 2017 at 23:12 by Martin Rubey
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Last Updated: Aug 11, 2017 at 23:12 by Martin Rubey