- FindStat

Identifier

St000938: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [2]=>0
[1,1]=>0
[3]=>0
[2,1]=>1
[1,1,1]=>0
[4]=>0
[3,1]=>1
[2,2]=>2
[2,1,1]=>1
[1,1,1,1]=>0
[5]=>0
[4,1]=>2
[3,2]=>1
[3,1,1]=>4
[2,2,1]=>1
[2,1,1,1]=>2
[1,1,1,1,1]=>0
[6]=>0
[5,1]=>2
[4,2]=>4
[4,1,1]=>3
[3,3]=>2
[3,2,1]=>7
[3,1,1,1]=>3
[2,2,2]=>2
[2,2,1,1]=>4
[2,1,1,1,1]=>2
[1,1,1,1,1,1]=>0
[7]=>0
[6,1]=>4
[5,2]=>5
[5,1,1]=>4
[4,3]=>4
[4,2,1]=>3
[4,1,1,1]=>9
[3,3,1]=>3
[3,2,2]=>3
[3,2,1,1]=>3
[3,1,1,1,1]=>4
[2,2,2,1]=>4
[2,2,1,1,1]=>5
[2,1,1,1,1,1]=>4
[1,1,1,1,1,1,1]=>0
[8]=>0
[7,1]=>4
[6,2]=>6
[6,1,1]=>7
[5,3]=>5
[5,2,1]=>12
[5,1,1,1]=>6
[4,4]=>6
[4,3,1]=>7
[4,2,2]=>9
[4,2,1,1]=>16
[4,1,1,1,1]=>6
[3,3,2]=>13
[3,3,1,1]=>9
[3,2,2,1]=>7
[3,2,1,1,1]=>12
[3,1,1,1,1,1]=>7
[2,2,2,2]=>6
[2,2,2,1,1]=>5
[2,2,1,1,1,1]=>6
[2,1,1,1,1,1,1]=>4
[1,1,1,1,1,1,1,1]=>0
[9]=>0
[8,1]=>7
[7,2]=>9
[7,1,1]=>8
[6,3]=>13
[6,2,1]=>8
[6,1,1,1]=>9
[5,4]=>10
[5,3,1]=>17
[5,2,2]=>16
[5,2,1,1]=>9
[5,1,1,1,1]=>19
[4,4,1]=>8
[4,3,2]=>10
[4,3,1,1]=>11
[4,2,2,1]=>11
[4,2,1,1,1]=>9
[4,1,1,1,1,1]=>9
[3,3,3]=>18
[3,3,2,1]=>10
[3,3,1,1,1]=>16
[3,2,2,2]=>8
[3,2,2,1,1]=>17
[3,2,1,1,1,1]=>8
[3,1,1,1,1,1,1]=>8
[2,2,2,2,1]=>10
[2,2,2,1,1,1]=>13
[2,2,1,1,1,1,1]=>9
[2,1,1,1,1,1,1,1]=>7
[1,1,1,1,1,1,1,1,1]=>0
[10]=>0
[9,1]=>8
[8,2]=>11
[8,1,1]=>12
[7,3]=>14
[7,2,1]=>19
[7,1,1,1]=>11
[6,4]=>11
[6,3,1]=>8
[6,2,2]=>15
[6,2,1,1]=>13
[6,1,1,1,1]=>14
[5,5]=>13
[5,4,1]=>20
[5,3,2]=>14
[5,3,1,1]=>23
[5,2,2,1]=>15
[5,2,1,1,1]=>30
[5,1,1,1,1,1]=>14
[4,4,2]=>13
[4,4,1,1]=>14
[4,3,3]=>14
[4,3,2,1]=>34
[4,3,1,1,1]=>15
[4,2,2,2]=>14
[4,2,2,1,1]=>23
[4,2,1,1,1,1]=>13
[4,1,1,1,1,1,1]=>11
[3,3,3,1]=>14
[3,3,2,2]=>13
[3,3,2,1,1]=>14
[3,3,1,1,1,1]=>15
[3,2,2,2,1]=>20
[3,2,2,1,1,1]=>8
[3,2,1,1,1,1,1]=>19
[3,1,1,1,1,1,1,1]=>12
[2,2,2,2,2]=>13
[2,2,2,2,1,1]=>11
[2,2,2,1,1,1,1]=>14
[2,2,1,1,1,1,1,1]=>11
[2,1,1,1,1,1,1,1,1]=>8
[1,1,1,1,1,1,1,1,1,1]=>0
[11]=>0
[10,1]=>12
[9,2]=>14
[9,1,1]=>15
[8,3]=>16
[8,2,1]=>15
[8,1,1,1]=>19
[7,4]=>7
[7,3,1]=>21
[7,2,2]=>17
[7,2,1,1]=>28
[7,1,1,1,1]=>15
[6,5]=>19
[6,4,1]=>17
[6,3,2]=>21
[6,3,1,1]=>30
[6,2,2,1]=>29
[6,2,1,1,1]=>15
[6,1,1,1,1,1]=>37
[5,5,1]=>14
[5,4,2]=>24
[5,4,1,1]=>12
[5,3,3]=>15
[5,3,2,1]=>18
[5,3,1,1,1]=>16
[5,2,2,2]=>25
[5,2,2,1,1]=>16
[5,2,1,1,1,1]=>15
[5,1,1,1,1,1,1]=>15
[4,4,3]=>16
[4,4,2,1]=>21
[4,4,1,1,1]=>25
[4,3,3,1]=>39
[4,3,2,2]=>21
[4,3,2,1,1]=>18
[4,3,1,1,1,1]=>29
[4,2,2,2,1]=>12
[4,2,2,1,1,1]=>30
[4,2,1,1,1,1,1]=>28
[4,1,1,1,1,1,1,1]=>19
[3,3,3,2]=>16
[3,3,3,1,1]=>15
[3,3,2,2,1]=>24
[3,3,2,1,1,1]=>21
[3,3,1,1,1,1,1]=>17
[3,2,2,2,2]=>14
[3,2,2,2,1,1]=>17
[3,2,2,1,1,1,1]=>21
[3,2,1,1,1,1,1,1]=>15
[3,1,1,1,1,1,1,1,1]=>15
[2,2,2,2,2,1]=>19
[2,2,2,2,1,1,1]=>7
[2,2,2,1,1,1,1,1]=>16
[2,2,1,1,1,1,1,1,1]=>14
[2,1,1,1,1,1,1,1,1,1]=>12
[1,1,1,1,1,1,1,1,1,1,1]=>0
[12]=>0
[11,1]=>14
[10,2]=>20
[10,1,1]=>18
[9,3]=>18
[9,2,1]=>31
[9,1,1,1]=>21
[8,4]=>23
[8,3,1]=>30
[8,2,2]=>24
[8,2,1,1]=>22
[8,1,1,1,1]=>19
[7,5]=>18
[7,4,1]=>42
[7,3,2]=>31
[7,3,1,1]=>27
[7,2,2,1]=>28
[7,2,1,1,1]=>38
[7,1,1,1,1,1]=>23
[6,6]=>26
[6,5,1]=>18
[6,4,2]=>46
[6,4,1,1]=>30
[6,3,3]=>27
[6,3,2,1]=>55
[6,3,1,1,1]=>41
[6,2,2,2]=>35
[6,2,2,1,1]=>41
[6,2,1,1,1,1]=>53
[6,1,1,1,1,1,1]=>23
[5,5,2]=>26
[5,5,1,1]=>29
[5,4,3]=>34
[5,4,2,1]=>34
[5,4,1,1,1]=>36
[5,3,3,1]=>26
[5,3,2,2]=>35
[5,3,2,1,1]=>53
[5,3,1,1,1,1]=>41
[5,2,2,2,1]=>36
[5,2,2,1,1,1]=>41
[5,2,1,1,1,1,1]=>38
[5,1,1,1,1,1,1,1]=>19
[4,4,4]=>24
[4,4,3,1]=>25
[4,4,2,2]=>54
[4,4,2,1,1]=>35
[4,4,1,1,1,1]=>35
[4,3,3,2]=>25
[4,3,3,1,1]=>26
[4,3,2,2,1]=>34
[4,3,2,1,1,1]=>55
[4,3,1,1,1,1,1]=>28
[4,2,2,2,2]=>29
[4,2,2,2,1,1]=>30
[4,2,2,1,1,1,1]=>27
[4,2,1,1,1,1,1,1]=>22
[4,1,1,1,1,1,1,1,1]=>21
[3,3,3,3]=>24
[3,3,3,2,1]=>34
[3,3,3,1,1,1]=>27
[3,3,2,2,2]=>26
[3,3,2,2,1,1]=>46
[3,3,2,1,1,1,1]=>31
[3,3,1,1,1,1,1,1]=>24
[3,2,2,2,2,1]=>18
[3,2,2,2,1,1,1]=>42
[3,2,2,1,1,1,1,1]=>30
[3,2,1,1,1,1,1,1,1]=>31
[3,1,1,1,1,1,1,1,1,1]=>18
[2,2,2,2,2,2]=>26
[2,2,2,2,2,1,1]=>18
[2,2,2,2,1,1,1,1]=>23
[2,2,2,1,1,1,1,1,1]=>18
[2,2,1,1,1,1,1,1,1,1]=>20
[2,1,1,1,1,1,1,1,1,1,1]=>14
[1,1,1,1,1,1,1,1,1,1,1,1]=>0
                    

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Description

The number of zeros 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 23:12 by Martin Rubey

Updated

Aug 11, 2017 at 23:12 by Martin Rubey