*****************************************************************************
* www.FindStat.org - The Combinatorial Statistic Finder *
* *
* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
* *
* This information is distributed in the hope that it will be useful, *
* but WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. *
*****************************************************************************
-----------------------------------------------------------------------------
Statistic identifier: St000936
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The number of even values 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 conjugace class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $4$.
It is shown in [1] that the sum of the values of the statistic over all partitions of a given size is even.
-----------------------------------------------------------------------------
References: [1] Miller, A. R. Note on parity and the irreducible characters of the symmetric group [[arXiv:1708.03267]]
-----------------------------------------------------------------------------
Code:
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 is_even(e)])
-----------------------------------------------------------------------------
Statistic values:
[2] => 0
[1,1] => 0
[3] => 0
[2,1] => 2
[1,1,1] => 0
[4] => 0
[3,1] => 1
[2,2] => 4
[2,1,1] => 1
[1,1,1,1] => 0
[5] => 0
[4,1] => 4
[3,2] => 1
[3,1,1] => 6
[2,2,1] => 1
[2,1,1,1] => 4
[1,1,1,1,1] => 0
[6] => 0
[5,1] => 3
[4,2] => 4
[4,1,1] => 7
[3,3] => 3
[3,2,1] => 10
[3,1,1,1] => 7
[2,2,2] => 3
[2,2,1,1] => 4
[2,1,1,1,1] => 3
[1,1,1,1,1,1] => 0
[7] => 0
[6,1] => 8
[5,2] => 11
[5,1,1] => 4
[4,3] => 9
[4,2,1] => 3
[4,1,1,1] => 14
[3,3,1] => 3
[3,2,2] => 3
[3,2,1,1] => 3
[3,1,1,1,1] => 4
[2,2,2,1] => 9
[2,2,1,1,1] => 11
[2,1,1,1,1,1] => 8
[1,1,1,1,1,1,1] => 0
[8] => 0
[7,1] => 7
[6,2] => 13
[6,1,1] => 9
[5,3] => 13
[5,2,1] => 18
[5,1,1,1] => 8
[4,4] => 15
[4,3,1] => 14
[4,2,2] => 15
[4,2,1,1] => 21
[4,1,1,1,1] => 8
[3,3,2] => 21
[3,3,1,1] => 15
[3,2,2,1] => 14
[3,2,1,1,1] => 18
[3,1,1,1,1,1] => 9
[2,2,2,2] => 15
[2,2,2,1,1] => 13
[2,2,1,1,1,1] => 13
[2,1,1,1,1,1,1] => 7
[1,1,1,1,1,1,1,1] => 0
[9] => 0
[8,1] => 15
[7,2] => 11
[7,1,1] => 19
[6,3] => 25
[6,2,1] => 8
[6,1,1,1] => 18
[5,4] => 20
[5,3,1] => 24
[5,2,2] => 22
[5,2,1,1] => 9
[5,1,1,1,1] => 29
[4,4,1] => 19
[4,3,2] => 18
[4,3,1,1] => 17
[4,2,2,1] => 17
[4,2,1,1,1] => 9
[4,1,1,1,1,1] => 18
[3,3,3] => 29
[3,3,2,1] => 18
[3,3,1,1,1] => 22
[3,2,2,2] => 19
[3,2,2,1,1] => 24
[3,2,1,1,1,1] => 8
[3,1,1,1,1,1,1] => 19
[2,2,2,2,1] => 20
[2,2,2,1,1,1] => 25
[2,2,1,1,1,1,1] => 11
[2,1,1,1,1,1,1,1] => 15
[1,1,1,1,1,1,1,1,1] => 0
[10] => 0
[9,1] => 15
[8,2] => 19
[8,1,1] => 23
[7,3] => 19
[7,2,1] => 32
[7,1,1,1] => 22
[6,4] => 25
[6,3,1] => 8
[6,2,2] => 19
[6,2,1,1] => 25
[6,1,1,1,1] => 28
[5,5] => 28
[5,4,1] => 33
[5,3,2] => 27
[5,3,1,1] => 24
[5,2,2,1] => 18
[5,2,1,1,1] => 41
[5,1,1,1,1,1] => 28
[4,4,2] => 26
[4,4,1,1] => 23
[4,3,3] => 28
[4,3,2,1] => 41
[4,3,1,1,1] => 18
[4,2,2,2] => 23
[4,2,2,1,1] => 24
[4,2,1,1,1,1] => 25
[4,1,1,1,1,1,1] => 22
[3,3,3,1] => 28
[3,3,2,2] => 26
[3,3,2,1,1] => 27
[3,3,1,1,1,1] => 19
[3,2,2,2,1] => 33
[3,2,2,1,1,1] => 8
[3,2,1,1,1,1,1] => 32
[3,1,1,1,1,1,1,1] => 23
[2,2,2,2,2] => 28
[2,2,2,2,1,1] => 25
[2,2,2,1,1,1,1] => 19
[2,2,1,1,1,1,1,1] => 19
[2,1,1,1,1,1,1,1,1] => 15
[1,1,1,1,1,1,1,1,1,1] => 0
[11] => 0
[10,1] => 27
[9,2] => 33
[9,1,1] => 23
[8,3] => 35
[8,2,1] => 18
[8,1,1,1] => 42
[7,4] => 8
[7,3,1] => 38
[7,2,2] => 22
[7,2,1,1] => 43
[7,1,1,1,1] => 35
[6,5] => 43
[6,4,1] => 22
[6,3,2] => 34
[6,3,1,1] => 46
[6,2,2,1] => 49
[6,2,1,1,1] => 32
[6,1,1,1,1,1] => 55
[5,5,1] => 35
[5,4,2] => 44
[5,4,1,1] => 16
[5,3,3] => 35
[5,3,2,1] => 34
[5,3,1,1,1] => 34
[5,2,2,2] => 27
[5,2,2,1,1] => 34
[5,2,1,1,1,1] => 32
[5,1,1,1,1,1,1] => 35
[4,4,3] => 38
[4,4,2,1] => 44
[4,4,1,1,1] => 27
[4,3,3,1] => 55
[4,3,2,2] => 44
[4,3,2,1,1] => 34
[4,3,1,1,1,1] => 49
[4,2,2,2,1] => 16
[4,2,2,1,1,1] => 46
[4,2,1,1,1,1,1] => 43
[4,1,1,1,1,1,1,1] => 42
[3,3,3,2] => 38
[3,3,3,1,1] => 35
[3,3,2,2,1] => 44
[3,3,2,1,1,1] => 34
[3,3,1,1,1,1,1] => 22
[3,2,2,2,2] => 35
[3,2,2,2,1,1] => 22
[3,2,2,1,1,1,1] => 38
[3,2,1,1,1,1,1,1] => 18
[3,1,1,1,1,1,1,1,1] => 23
[2,2,2,2,2,1] => 43
[2,2,2,2,1,1,1] => 8
[2,2,2,1,1,1,1,1] => 35
[2,2,1,1,1,1,1,1,1] => 33
[2,1,1,1,1,1,1,1,1,1] => 27
[1,1,1,1,1,1,1,1,1,1,1] => 0
[12] => 0
[11,1] => 29
[10,2] => 42
[10,1,1] => 35
[9,3] => 42
[9,2,1] => 55
[9,1,1,1] => 36
[8,4] => 32
[8,3,1] => 37
[8,2,2] => 44
[8,2,1,1] => 22
[8,1,1,1,1] => 44
[7,5] => 24
[7,4,1] => 71
[7,3,2] => 40
[7,3,1,1] => 46
[7,2,2,1] => 39
[7,2,1,1,1] => 59
[7,1,1,1,1,1] => 49
[6,6] => 60
[6,5,1] => 23
[6,4,2] => 50
[6,4,1,1] => 50
[6,3,3] => 45
[6,3,2,1] => 73
[6,3,1,1,1] => 67
[6,2,2,2] => 44
[6,2,2,1,1] => 62
[6,2,1,1,1,1] => 76
[6,1,1,1,1,1,1] => 49
[5,5,2] => 48
[5,5,1,1] => 40
[5,4,3] => 59
[5,4,2,1] => 41
[5,4,1,1,1] => 57
[5,3,3,1] => 49
[5,3,2,2] => 44
[5,3,2,1,1] => 76
[5,3,1,1,1,1] => 62
[5,2,2,2,1] => 57
[5,2,2,1,1,1] => 67
[5,2,1,1,1,1,1] => 59
[5,1,1,1,1,1,1,1] => 44
[4,4,4] => 55
[4,4,3,1] => 47
[4,4,2,2] => 76
[4,4,2,1,1] => 44
[4,4,1,1,1,1] => 44
[4,3,3,2] => 47
[4,3,3,1,1] => 49
[4,3,2,2,1] => 41
[4,3,2,1,1,1] => 73
[4,3,1,1,1,1,1] => 39
[4,2,2,2,2] => 40
[4,2,2,2,1,1] => 50
[4,2,2,1,1,1,1] => 46
[4,2,1,1,1,1,1,1] => 22
[4,1,1,1,1,1,1,1,1] => 36
[3,3,3,3] => 55
[3,3,3,2,1] => 59
[3,3,3,1,1,1] => 45
[3,3,2,2,2] => 48
[3,3,2,2,1,1] => 50
[3,3,2,1,1,1,1] => 40
[3,3,1,1,1,1,1,1] => 44
[3,2,2,2,2,1] => 23
[3,2,2,2,1,1,1] => 71
[3,2,2,1,1,1,1,1] => 37
[3,2,1,1,1,1,1,1,1] => 55
[3,1,1,1,1,1,1,1,1,1] => 35
[2,2,2,2,2,2] => 60
[2,2,2,2,2,1,1] => 24
[2,2,2,2,1,1,1,1] => 32
[2,2,2,1,1,1,1,1,1] => 42
[2,2,1,1,1,1,1,1,1,1] => 42
[2,1,1,1,1,1,1,1,1,1,1] => 29
[1,1,1,1,1,1,1,1,1,1,1,1] => 0
-----------------------------------------------------------------------------
Created: Aug 11, 2017 at 22:54 by Martin Rubey
-----------------------------------------------------------------------------
Last Updated: Aug 11, 2017 at 22:54 by Martin Rubey