Identifier
Identifier
Values
[2] generating graphics... => 0
[1,1] generating graphics... => 0
[3] generating graphics... => 0
[2,1] generating graphics... => 2
[1,1,1] generating graphics... => 0
[4] generating graphics... => 0
[3,1] generating graphics... => 1
[2,2] generating graphics... => 4
[2,1,1] generating graphics... => 1
[1,1,1,1] generating graphics... => 0
[5] generating graphics... => 0
[4,1] generating graphics... => 4
[3,2] generating graphics... => 1
[3,1,1] generating graphics... => 6
[2,2,1] generating graphics... => 1
[2,1,1,1] generating graphics... => 4
[1,1,1,1,1] generating graphics... => 0
[6] generating graphics... => 0
[5,1] generating graphics... => 3
[4,2] generating graphics... => 4
[4,1,1] generating graphics... => 7
[3,3] generating graphics... => 3
[3,2,1] generating graphics... => 10
[3,1,1,1] generating graphics... => 7
[2,2,2] generating graphics... => 3
[2,2,1,1] generating graphics... => 4
[2,1,1,1,1] generating graphics... => 3
[1,1,1,1,1,1] generating graphics... => 0
[7] generating graphics... => 0
[6,1] generating graphics... => 8
[5,2] generating graphics... => 11
[5,1,1] generating graphics... => 4
[4,3] generating graphics... => 9
[4,2,1] generating graphics... => 3
[4,1,1,1] generating graphics... => 14
[3,3,1] generating graphics... => 3
[3,2,2] generating graphics... => 3
[3,2,1,1] generating graphics... => 3
[3,1,1,1,1] generating graphics... => 4
[2,2,2,1] generating graphics... => 9
[2,2,1,1,1] generating graphics... => 11
[2,1,1,1,1,1] generating graphics... => 8
[1,1,1,1,1,1,1] generating graphics... => 0
[8] generating graphics... => 0
[7,1] generating graphics... => 7
[6,2] generating graphics... => 13
[6,1,1] generating graphics... => 9
[5,3] generating graphics... => 13
[5,2,1] generating graphics... => 18
[5,1,1,1] generating graphics... => 8
[4,4] generating graphics... => 15
[4,3,1] generating graphics... => 14
[4,2,2] generating graphics... => 15
[4,2,1,1] generating graphics... => 21
[4,1,1,1,1] generating graphics... => 8
[3,3,2] generating graphics... => 21
[3,3,1,1] generating graphics... => 15
[3,2,2,1] generating graphics... => 14
[3,2,1,1,1] generating graphics... => 18
[3,1,1,1,1,1] generating graphics... => 9
[2,2,2,2] generating graphics... => 15
[2,2,2,1,1] generating graphics... => 13
[2,2,1,1,1,1] generating graphics... => 13
[2,1,1,1,1,1,1] generating graphics... => 7
[1,1,1,1,1,1,1,1] generating graphics... => 0
[9] generating graphics... => 0
[8,1] generating graphics... => 15
[7,2] generating graphics... => 11
[7,1,1] generating graphics... => 19
[6,3] generating graphics... => 25
[6,2,1] generating graphics... => 8
[6,1,1,1] generating graphics... => 18
[5,4] generating graphics... => 20
[5,3,1] generating graphics... => 24
[5,2,2] generating graphics... => 22
[5,2,1,1] generating graphics... => 9
[5,1,1,1,1] generating graphics... => 29
[4,4,1] generating graphics... => 19
[4,3,2] generating graphics... => 18
[4,3,1,1] generating graphics... => 17
[4,2,2,1] generating graphics... => 17
[4,2,1,1,1] generating graphics... => 9
[4,1,1,1,1,1] generating graphics... => 18
[3,3,3] generating graphics... => 29
[3,3,2,1] generating graphics... => 18
[3,3,1,1,1] generating graphics... => 22
[3,2,2,2] generating graphics... => 19
[3,2,2,1,1] generating graphics... => 24
[3,2,1,1,1,1] generating graphics... => 8
[3,1,1,1,1,1,1] generating graphics... => 19
[2,2,2,2,1] generating graphics... => 20
[2,2,2,1,1,1] generating graphics... => 25
[2,2,1,1,1,1,1] generating graphics... => 11
[2,1,1,1,1,1,1,1] generating graphics... => 15
[1,1,1,1,1,1,1,1,1] generating graphics... => 0
[10] generating graphics... => 0
[9,1] generating graphics... => 15
[8,2] generating graphics... => 19
[8,1,1] generating graphics... => 23
[7,3] generating graphics... => 19
[7,2,1] generating graphics... => 32
[7,1,1,1] generating graphics... => 22
[6,4] generating graphics... => 25
[6,3,1] generating graphics... => 8
[6,2,2] generating graphics... => 19
[6,2,1,1] generating graphics... => 25
[6,1,1,1,1] generating graphics... => 28
[5,5] generating graphics... => 28
[5,4,1] generating graphics... => 33
[5,3,2] generating graphics... => 27
[5,3,1,1] generating graphics... => 24
[5,2,2,1] generating graphics... => 18
[5,2,1,1,1] generating graphics... => 41
[5,1,1,1,1,1] generating graphics... => 28
[4,4,2] generating graphics... => 26
[4,4,1,1] generating graphics... => 23
[4,3,3] generating graphics... => 28
[4,3,2,1] generating graphics... => 41
[4,3,1,1,1] generating graphics... => 18
[4,2,2,2] generating graphics... => 23
[4,2,2,1,1] generating graphics... => 24
[4,2,1,1,1,1] generating graphics... => 25
[4,1,1,1,1,1,1] generating graphics... => 22
[3,3,3,1] generating graphics... => 28
[3,3,2,2] generating graphics... => 26
[3,3,2,1,1] generating graphics... => 27
[3,3,1,1,1,1] generating graphics... => 19
[3,2,2,2,1] generating graphics... => 33
[3,2,2,1,1,1] generating graphics... => 8
[3,2,1,1,1,1,1] generating graphics... => 32
[3,1,1,1,1,1,1,1] generating graphics... => 23
[2,2,2,2,2] generating graphics... => 28
[2,2,2,2,1,1] generating graphics... => 25
[2,2,2,1,1,1,1] generating graphics... => 19
[2,2,1,1,1,1,1,1] generating graphics... => 19
[2,1,1,1,1,1,1,1,1] generating graphics... => 15
[1,1,1,1,1,1,1,1,1,1] generating graphics... => 0
[11] generating graphics... => 0
[10,1] generating graphics... => 27
[9,2] generating graphics... => 33
[9,1,1] generating graphics... => 23
[8,3] generating graphics... => 35
[8,2,1] generating graphics... => 18
[8,1,1,1] generating graphics... => 42
[7,4] generating graphics... => 8
[7,3,1] generating graphics... => 38
[7,2,2] generating graphics... => 22
[7,2,1,1] generating graphics... => 43
[7,1,1,1,1] generating graphics... => 35
[6,5] generating graphics... => 43
[6,4,1] generating graphics... => 22
[6,3,2] generating graphics... => 34
[6,3,1,1] generating graphics... => 46
[6,2,2,1] generating graphics... => 49
[6,2,1,1,1] generating graphics... => 32
[6,1,1,1,1,1] generating graphics... => 55
[5,5,1] generating graphics... => 35
[5,4,2] generating graphics... => 44
[5,4,1,1] generating graphics... => 16
[5,3,3] generating graphics... => 35
[5,3,2,1] generating graphics... => 34
[5,3,1,1,1] generating graphics... => 34
[5,2,2,2] generating graphics... => 27
[5,2,2,1,1] generating graphics... => 34
[5,2,1,1,1,1] generating graphics... => 32
[5,1,1,1,1,1,1] generating graphics... => 35
[4,4,3] generating graphics... => 38
[4,4,2,1] generating graphics... => 44
[4,4,1,1,1] generating graphics... => 27
[4,3,3,1] generating graphics... => 55
[4,3,2,2] generating graphics... => 44
[4,3,2,1,1] generating graphics... => 34
[4,3,1,1,1,1] generating graphics... => 49
[4,2,2,2,1] generating graphics... => 16
[4,2,2,1,1,1] generating graphics... => 46
[4,2,1,1,1,1,1] generating graphics... => 43
[4,1,1,1,1,1,1,1] generating graphics... => 42
[3,3,3,2] generating graphics... => 38
[3,3,3,1,1] generating graphics... => 35
[3,3,2,2,1] generating graphics... => 44
[3,3,2,1,1,1] generating graphics... => 34
[3,3,1,1,1,1,1] generating graphics... => 22
[3,2,2,2,2] generating graphics... => 35
[3,2,2,2,1,1] generating graphics... => 22
[3,2,2,1,1,1,1] generating graphics... => 38
[3,2,1,1,1,1,1,1] generating graphics... => 18
[3,1,1,1,1,1,1,1,1] generating graphics... => 23
[2,2,2,2,2,1] generating graphics... => 43
[2,2,2,2,1,1,1] generating graphics... => 8
[2,2,2,1,1,1,1,1] generating graphics... => 35
[2,2,1,1,1,1,1,1,1] generating graphics... => 33
[2,1,1,1,1,1,1,1,1,1] generating graphics... => 27
[1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 0
[12] generating graphics... => 0
[11,1] generating graphics... => 29
[10,2] generating graphics... => 42
[10,1,1] generating graphics... => 35
[9,3] generating graphics... => 42
[9,2,1] generating graphics... => 55
[9,1,1,1] generating graphics... => 36
[8,4] generating graphics... => 32
[8,3,1] generating graphics... => 37
[8,2,2] generating graphics... => 44
[8,2,1,1] generating graphics... => 22
[8,1,1,1,1] generating graphics... => 44
[7,5] generating graphics... => 24
[7,4,1] generating graphics... => 71
[7,3,2] generating graphics... => 40
[7,3,1,1] generating graphics... => 46
[7,2,2,1] generating graphics... => 39
[7,2,1,1,1] generating graphics... => 59
[7,1,1,1,1,1] generating graphics... => 49
[6,6] generating graphics... => 60
[6,5,1] generating graphics... => 23
[6,4,2] generating graphics... => 50
[6,4,1,1] generating graphics... => 50
[6,3,3] generating graphics... => 45
[6,3,2,1] generating graphics... => 73
[6,3,1,1,1] generating graphics... => 67
[6,2,2,2] generating graphics... => 44
[6,2,2,1,1] generating graphics... => 62
[6,2,1,1,1,1] generating graphics... => 76
[6,1,1,1,1,1,1] generating graphics... => 49
[5,5,2] generating graphics... => 48
[5,5,1,1] generating graphics... => 40
[5,4,3] generating graphics... => 59
[5,4,2,1] generating graphics... => 41
[5,4,1,1,1] generating graphics... => 57
[5,3,3,1] generating graphics... => 49
[5,3,2,2] generating graphics... => 44
[5,3,2,1,1] generating graphics... => 76
[5,3,1,1,1,1] generating graphics... => 62
[5,2,2,2,1] generating graphics... => 57
[5,2,2,1,1,1] generating graphics... => 67
[5,2,1,1,1,1,1] generating graphics... => 59
[5,1,1,1,1,1,1,1] generating graphics... => 44
[4,4,4] generating graphics... => 55
[4,4,3,1] generating graphics... => 47
[4,4,2,2] generating graphics... => 76
[4,4,2,1,1] generating graphics... => 44
[4,4,1,1,1,1] generating graphics... => 44
[4,3,3,2] generating graphics... => 47
[4,3,3,1,1] generating graphics... => 49
[4,3,2,2,1] generating graphics... => 41
[4,3,2,1,1,1] generating graphics... => 73
[4,3,1,1,1,1,1] generating graphics... => 39
[4,2,2,2,2] generating graphics... => 40
[4,2,2,2,1,1] generating graphics... => 50
[4,2,2,1,1,1,1] generating graphics... => 46
[4,2,1,1,1,1,1,1] generating graphics... => 22
[4,1,1,1,1,1,1,1,1] generating graphics... => 36
[3,3,3,3] generating graphics... => 55
[3,3,3,2,1] generating graphics... => 59
[3,3,3,1,1,1] generating graphics... => 45
[3,3,2,2,2] generating graphics... => 48
[3,3,2,2,1,1] generating graphics... => 50
[3,3,2,1,1,1,1] generating graphics... => 40
[3,3,1,1,1,1,1,1] generating graphics... => 44
[3,2,2,2,2,1] generating graphics... => 23
[3,2,2,2,1,1,1] generating graphics... => 71
[3,2,2,1,1,1,1,1] generating graphics... => 37
[3,2,1,1,1,1,1,1,1] generating graphics... => 55
[3,1,1,1,1,1,1,1,1,1] generating graphics... => 35
[2,2,2,2,2,2] generating graphics... => 60
[2,2,2,2,2,1,1] generating graphics... => 24
[2,2,2,2,1,1,1,1] generating graphics... => 32
[2,2,2,1,1,1,1,1,1] generating graphics... => 42
[2,2,1,1,1,1,1,1,1,1] generating graphics... => 42
[2,1,1,1,1,1,1,1,1,1,1] generating graphics... => 29
[1,1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 0
click to show generating function       
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)])

Created
Aug 11, 2017 at 22:54 by Martin Rubey
Updated
Aug 11, 2017 at 22:54 by Martin Rubey