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-----------------------------------------------------------------------------
Statistic identifier: St000940
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The number of characters of the symmetric group whose value on the partition is zero.
The maximal value for any given size is recorded in [2].
-----------------------------------------------------------------------------
References: [1] Miller, A. R. Note on parity and the irreducible characters of the symmetric group [[arXiv:1708.03267]]
[2] Maximal number of zeros in a column of the character table of the symmetric group S_n. [[OEIS:A086642]]
-----------------------------------------------------------------------------
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 is_even(e)])
-----------------------------------------------------------------------------
Statistic values:
[2] => 0
[1,1] => 0
[3] => 0
[2,1] => 1
[1,1,1] => 0
[4] => 1
[3,1] => 2
[2,2] => 0
[2,1,1] => 1
[1,1,1,1] => 0
[5] => 2
[4,1] => 3
[3,2] => 1
[3,1,1] => 1
[2,2,1] => 2
[2,1,1,1] => 1
[1,1,1,1,1] => 0
[6] => 5
[5,1] => 6
[4,2] => 3
[4,1,1] => 3
[3,3] => 2
[3,2,1] => 5
[3,1,1,1] => 2
[2,2,2] => 1
[2,2,1,1] => 1
[2,1,1,1,1] => 1
[1,1,1,1,1,1] => 0
[7] => 8
[6,1] => 9
[5,2] => 5
[5,1,1] => 5
[4,3] => 3
[4,2,1] => 7
[4,1,1,1] => 3
[3,3,1] => 6
[3,2,2] => 0
[3,2,1,1] => 3
[3,1,1,1,1] => 0
[2,2,2,1] => 5
[2,2,1,1,1] => 0
[2,1,1,1,1,1] => 1
[1,1,1,1,1,1,1] => 0
[8] => 14
[7,1] => 15
[6,2] => 10
[6,1,1] => 10
[5,3] => 7
[5,2,1] => 12
[5,1,1,1] => 7
[4,4] => 8
[4,3,1] => 10
[4,2,2] => 8
[4,2,1,1] => 12
[4,1,1,1,1] => 6
[3,3,2] => 4
[3,3,1,1] => 4
[3,2,2,1] => 7
[3,2,1,1,1] => 4
[3,1,1,1,1,1] => 1
[2,2,2,2] => 2
[2,2,2,1,1] => 6
[2,2,1,1,1,1] => 4
[2,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1] => 0
[9] => 21
[8,1] => 22
[7,2] => 16
[7,1,1] => 16
[6,3] => 12
[6,2,1] => 18
[6,1,1,1] => 12
[5,4] => 10
[5,3,1] => 15
[5,2,2] => 5
[5,2,1,1] => 10
[5,1,1,1,1] => 5
[4,4,1] => 16
[4,3,2] => 12
[4,3,1,1] => 12
[4,2,2,1] => 16
[4,2,1,1,1] => 12
[4,1,1,1,1,1] => 6
[3,3,3] => 8
[3,3,2,1] => 12
[3,3,1,1,1] => 13
[3,2,2,2] => 4
[3,2,2,1,1] => 6
[3,2,1,1,1,1] => 4
[3,1,1,1,1,1,1] => 6
[2,2,2,2,1] => 10
[2,2,2,1,1,1] => 4
[2,2,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1] => 0
[10] => 32
[9,1] => 33
[8,2] => 26
[8,1,1] => 26
[7,3] => 21
[7,2,1] => 28
[7,1,1,1] => 21
[6,4] => 18
[6,3,1] => 24
[6,2,2] => 12
[6,2,1,1] => 18
[6,1,1,1,1] => 12
[5,5] => 22
[5,4,1] => 22
[5,3,2] => 18
[5,3,1,1] => 18
[5,2,2,1] => 17
[5,2,1,1,1] => 16
[5,1,1,1,1,1] => 12
[4,4,2] => 14
[4,4,1,1] => 14
[4,3,3] => 6
[4,3,2,1] => 18
[4,3,1,1,1] => 6
[4,2,2,2] => 10
[4,2,2,1,1] => 14
[4,2,1,1,1,1] => 10
[4,1,1,1,1,1,1] => 2
[3,3,3,1] => 20
[3,3,2,2] => 7
[3,3,2,1,1] => 6
[3,3,1,1,1,1] => 7
[3,2,2,2,1] => 12
[3,2,2,1,1,1] => 7
[3,2,1,1,1,1,1] => 4
[3,1,1,1,1,1,1,1] => 6
[2,2,2,2,2] => 6
[2,2,2,2,1,1] => 6
[2,2,2,1,1,1,1] => 10
[2,2,1,1,1,1,1,1] => 5
[2,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1,1] => 0
[11] => 45
[10,1] => 46
[9,2] => 38
[9,1,1] => 38
[8,3] => 32
[8,2,1] => 40
[8,1,1,1] => 32
[7,4] => 28
[7,3,1] => 35
[7,2,2] => 21
[7,2,1,1] => 28
[7,1,1,1,1] => 21
[6,5] => 26
[6,4,1] => 32
[6,3,2] => 20
[6,3,1,1] => 20
[6,2,2,1] => 26
[6,2,1,1,1] => 20
[6,1,1,1,1,1] => 14
[5,5,1] => 36
[5,4,2] => 22
[5,4,1,1] => 22
[5,3,3] => 11
[5,3,2,1] => 26
[5,3,1,1,1] => 11
[5,2,2,2] => 10
[5,2,2,1,1] => 12
[5,2,1,1,1,1] => 10
[5,1,1,1,1,1,1] => 6
[4,4,3] => 14
[4,4,2,1] => 28
[4,4,1,1,1] => 14
[4,3,3,1] => 20
[4,3,2,2] => 14
[4,3,2,1,1] => 26
[4,3,1,1,1,1] => 8
[4,2,2,2,1] => 24
[4,2,2,1,1,1] => 14
[4,2,1,1,1,1,1] => 14
[4,1,1,1,1,1,1,1] => 4
[3,3,3,2] => 12
[3,3,3,1,1] => 12
[3,3,2,2,1] => 11
[3,3,2,1,1,1] => 22
[3,3,1,1,1,1,1] => 5
[3,2,2,2,2] => 0
[3,2,2,2,1,1] => 8
[3,2,2,1,1,1,1] => 2
[3,2,1,1,1,1,1,1] => 10
[3,1,1,1,1,1,1,1,1] => 0
[2,2,2,2,2,1] => 20
[2,2,2,2,1,1,1] => 0
[2,2,2,1,1,1,1,1] => 4
[2,2,1,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1,1,1] => 0
[12] => 65
[11,1] => 66
[10,2] => 57
[10,1,1] => 57
[9,3] => 50
[9,2,1] => 59
[9,1,1,1] => 50
[8,4] => 45
[8,3,1] => 53
[8,2,2] => 37
[8,2,1,1] => 45
[8,1,1,1,1] => 37
[7,5] => 42
[7,4,1] => 49
[7,3,2] => 35
[7,3,1,1] => 35
[7,2,2,1] => 42
[7,2,1,1,1] => 35
[7,1,1,1,1,1] => 28
[6,6] => 50
[6,5,1] => 47
[6,4,2] => 35
[6,4,1,1] => 35
[6,3,3] => 41
[6,3,2,1] => 47
[6,3,1,1,1] => 37
[6,2,2,2] => 26
[6,2,2,1,1] => 31
[6,2,1,1,1,1] => 26
[6,1,1,1,1,1,1] => 23
[5,5,2] => 37
[5,5,1,1] => 37
[5,4,3] => 29
[5,4,2,1] => 37
[5,4,1,1,1] => 25
[5,3,3,1] => 32
[5,3,2,2] => 18
[5,3,2,1,1] => 29
[5,3,1,1,1,1] => 14
[5,2,2,2,1] => 27
[5,2,2,1,1,1] => 16
[5,2,1,1,1,1,1] => 17
[5,1,1,1,1,1,1,1] => 12
[4,4,4] => 37
[4,4,3,1] => 35
[4,4,2,2] => 23
[4,4,2,1,1] => 37
[4,4,1,1,1,1] => 23
[4,3,3,2] => 19
[4,3,3,1,1] => 19
[4,3,2,2,1] => 39
[4,3,2,1,1,1] => 23
[4,3,1,1,1,1,1] => 15
[4,2,2,2,2] => 19
[4,2,2,2,1,1] => 19
[4,2,2,1,1,1,1] => 27
[4,2,1,1,1,1,1,1] => 11
[4,1,1,1,1,1,1,1,1] => 9
[3,3,3,3] => 26
[3,3,3,2,1] => 37
[3,3,3,1,1,1] => 26
[3,3,2,2,2] => 13
[3,3,2,2,1,1] => 12
[3,3,2,1,1,1,1] => 17
[3,3,1,1,1,1,1,1] => 8
[3,2,2,2,2,1] => 22
[3,2,2,2,1,1,1] => 17
[3,2,2,1,1,1,1,1] => 8
[3,2,1,1,1,1,1,1,1] => 15
[3,1,1,1,1,1,1,1,1,1] => 2
[2,2,2,2,2,2] => 12
[2,2,2,2,2,1,1] => 21
[2,2,2,2,1,1,1,1] => 12
[2,2,2,1,1,1,1,1,1] => 7
[2,2,1,1,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1,1,1] => 3
[1,1,1,1,1,1,1,1,1,1,1,1] => 0
-----------------------------------------------------------------------------
Created: Aug 12, 2017 at 11:37 by Martin Rubey
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Last Updated: Aug 12, 2017 at 13:44 by Martin Rubey