- FindStat

Identifier

St000936: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [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
                    

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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