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Identifier
Values
=>
Cc0002;cc-rep
[2]=>2 [1,1]=>1 [3]=>3 [2,1]=>1 [1,1,1]=>2 [4]=>5 [3,1]=>2 [2,2]=>2 [2,1,1]=>2 [1,1,1,1]=>3 [5]=>7 [4,1]=>3 [3,2]=>4 [3,1,1]=>2 [2,2,1]=>3 [2,1,1,1]=>3 [1,1,1,1,1]=>4 [6]=>11 [5,1]=>5 [4,2]=>5 [4,1,1]=>5 [3,3]=>5 [3,2,1]=>2 [3,1,1,1]=>5 [2,2,2]=>5 [2,2,1,1]=>4 [2,1,1,1,1]=>5 [1,1,1,1,1,1]=>6 [7]=>15 [6,1]=>7 [5,2]=>7 [5,1,1]=>6 [4,3]=>6 [4,2,1]=>5 [4,1,1,1]=>4 [3,3,1]=>7 [3,2,2]=>7 [3,2,1,1]=>5 [3,1,1,1,1]=>5 [2,2,2,1]=>5 [2,2,1,1,1]=>5 [2,1,1,1,1,1]=>6 [1,1,1,1,1,1,1]=>8 [8]=>22 [7,1]=>11 [6,2]=>12 [6,1,1]=>9 [5,3]=>11 [5,2,1]=>5 [5,1,1,1]=>10 [4,4]=>9 [4,3,1]=>8 [4,2,2]=>8 [4,2,1,1]=>3 [4,1,1,1,1]=>8 [3,3,2]=>5 [3,3,1,1]=>6 [3,2,2,1]=>7 [3,2,1,1,1]=>5 [3,1,1,1,1,1]=>9 [2,2,2,2]=>8 [2,2,2,1,1]=>9 [2,2,1,1,1,1]=>8 [2,1,1,1,1,1,1]=>9 [1,1,1,1,1,1,1,1]=>12 [9]=>30 [8,1]=>15 [7,2]=>16 [7,1,1]=>13 [6,3]=>11 [6,2,1]=>10 [6,1,1,1]=>13 [5,4]=>12 [5,3,1]=>7 [5,2,2]=>9 [5,2,1,1]=>12 [5,1,1,1,1]=>6 [4,4,1]=>11 [4,3,2]=>10 [4,3,1,1]=>10 [4,2,2,1]=>9 [4,2,1,1,1]=>10 [4,1,1,1,1,1]=>12 [3,3,3]=>7 [3,3,2,1]=>11 [3,3,1,1,1]=>7 [3,2,2,2]=>12 [3,2,2,1,1]=>7 [3,2,1,1,1,1]=>13 [3,1,1,1,1,1,1]=>12 [2,2,2,2,1]=>11 [2,2,2,1,1,1]=>9 [2,2,1,1,1,1,1]=>11 [2,1,1,1,1,1,1,1]=>12 [1,1,1,1,1,1,1,1,1]=>16 [10]=>42 [9,1]=>22 [8,2]=>24 [8,1,1]=>17 [7,3]=>18 [7,2,1]=>11 [7,1,1,1]=>21 [6,4]=>19 [6,3,1]=>16 [6,2,2]=>17 [6,2,1,1]=>13 [6,1,1,1,1]=>16 [5,5]=>15 [5,4,1]=>11 [5,3,2]=>14 [5,3,1,1]=>9 [5,2,2,1]=>14 [5,2,1,1,1]=>6 [5,1,1,1,1,1]=>16 [4,4,2]=>16 [4,4,1,1]=>15 [4,3,3]=>14 [4,3,2,1]=>4 [4,3,1,1,1]=>13 [4,2,2,2]=>15 [4,2,2,1,1]=>10 [4,2,1,1,1,1]=>13 [4,1,1,1,1,1,1]=>13 [3,3,3,1]=>16 [3,3,2,2]=>15 [3,3,2,1,1]=>15 [3,3,1,1,1,1]=>14 [3,2,2,2,1]=>11 [3,2,2,1,1,1]=>17 [3,2,1,1,1,1,1]=>11 [3,1,1,1,1,1,1,1]=>16 [2,2,2,2,2]=>17 [2,2,2,2,1,1]=>16 [2,2,2,1,1,1,1]=>15 [2,2,1,1,1,1,1,1]=>15 [2,1,1,1,1,1,1,1,1]=>18 [1,1,1,1,1,1,1,1,1,1]=>22 [11]=>56 [10,1]=>30 [9,2]=>33 [9,1,1]=>24 [8,3]=>27 [8,2,1]=>19 [8,1,1,1]=>25 [7,4]=>33 [7,3,1]=>18 [7,2,2]=>24 [7,2,1,1]=>15 [7,1,1,1,1]=>23 [6,5]=>21 [6,4,1]=>20 [6,3,2]=>18 [6,3,1,1]=>13 [6,2,2,1]=>14 [6,2,1,1,1]=>19 [6,1,1,1,1,1]=>12 [5,5,1]=>22 [5,4,2]=>17 [5,4,1,1]=>24 [5,3,3]=>21 [5,3,2,1]=>20 [5,3,1,1,1]=>21 [5,2,2,2]=>16 [5,2,2,1,1]=>21 [5,2,1,1,1,1]=>22 [5,1,1,1,1,1,1]=>20 [4,4,3]=>22 [4,4,2,1]=>19 [4,4,1,1,1]=>15 [4,3,3,1]=>9 [4,3,2,2]=>19 [4,3,2,1,1]=>18 [4,3,1,1,1,1]=>14 [4,2,2,2,1]=>20 [4,2,2,1,1,1]=>14 [4,2,1,1,1,1,1]=>14 [4,1,1,1,1,1,1,1]=>21 [3,3,3,2]=>20 [3,3,3,1,1]=>20 [3,3,2,2,1]=>14 [3,3,2,1,1,1]=>18 [3,3,1,1,1,1,1]=>19 [3,2,2,2,2]=>22 [3,2,2,2,1,1]=>21 [3,2,2,1,1,1,1]=>17 [3,2,1,1,1,1,1,1]=>20 [3,1,1,1,1,1,1,1,1]=>22 [2,2,2,2,2,1]=>20 [2,2,2,2,1,1,1]=>25 [2,2,2,1,1,1,1,1]=>21 [2,2,1,1,1,1,1,1,1]=>21 [2,1,1,1,1,1,1,1,1,1]=>23 [1,1,1,1,1,1,1,1,1,1,1]=>29 [12]=>77 [11,1]=>42 [10,2]=>46 [10,1,1]=>34 [9,3]=>40 [9,2,1]=>22 [9,1,1,1]=>37 [8,4]=>36 [8,3,1]=>25 [8,2,2]=>33 [8,2,1,1]=>31 [8,1,1,1,1]=>31 [7,5]=>35 [7,4,1]=>19 [7,3,2]=>24 [7,3,1,1]=>29 [7,2,2,1]=>24 [7,2,1,1,1]=>20 [7,1,1,1,1,1]=>31 [6,6]=>29 [6,5,1]=>30 [6,4,2]=>17 [6,4,1,1]=>27 [6,3,3]=>24 [6,3,2,1]=>11 [6,3,1,1,1]=>20 [6,2,2,2]=>24 [6,2,2,1,1]=>19 [6,2,1,1,1,1]=>13 [6,1,1,1,1,1,1]=>26 [5,5,2]=>26 [5,5,1,1]=>26 [5,4,3]=>23 [5,4,2,1]=>22 [5,4,1,1,1]=>19 [5,3,3,1]=>27 [5,3,2,2]=>21 [5,3,2,1,1]=>12 [5,3,1,1,1,1]=>19 [5,2,2,2,1]=>20 [5,2,2,1,1,1]=>18 [5,2,1,1,1,1,1]=>20 [5,1,1,1,1,1,1,1]=>29 [4,4,4]=>30 [4,4,3,1]=>26 [4,4,2,2]=>13 [4,4,2,1,1]=>19 [4,4,1,1,1,1]=>20 [4,3,3,2]=>26 [4,3,3,1,1]=>25 [4,3,2,2,1]=>22 [4,3,2,1,1,1]=>11 [4,3,1,1,1,1,1]=>24 [4,2,2,2,2]=>25 [4,2,2,2,1,1]=>26 [4,2,2,1,1,1,1]=>25 [4,2,1,1,1,1,1,1]=>30 [4,1,1,1,1,1,1,1,1]=>30 [3,3,3,3]=>27 [3,3,3,2,1]=>23 [3,3,3,1,1,1]=>25 [3,3,2,2,2]=>23 [3,3,2,2,1,1]=>17 [3,3,2,1,1,1,1]=>22 [3,3,1,1,1,1,1,1]=>28 [3,2,2,2,2,1]=>31 [3,2,2,2,1,1,1]=>19 [3,2,2,1,1,1,1,1]=>24 [3,2,1,1,1,1,1,1,1]=>23 [3,1,1,1,1,1,1,1,1,1]=>30 [2,2,2,2,2,2]=>29 [2,2,2,2,2,1,1]=>31 [2,2,2,2,1,1,1,1]=>27 [2,2,2,1,1,1,1,1,1]=>30 [2,2,1,1,1,1,1,1,1,1]=>29 [2,1,1,1,1,1,1,1,1,1,1]=>32 [1,1,1,1,1,1,1,1,1,1,1,1]=>40
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Description
The number of positive 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 conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
References
[1] Miller, A. R. Note on parity and the irreducible characters of the symmetric group arXiv:1708.03267
Code
@cached_function
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 e > 0])

Created
Aug 11, 2017 at 23:10 by Martin Rubey
Updated
Aug 11, 2017 at 23:10 by Martin Rubey