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Identifier
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