- FindStat

Identifier

St000937: Integer partitions ⟶ ℤ

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