- FindStat

Identifier

St000940: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [2]=>0
[1,1]=>0
[3]=>0
[2,1]=>1
[1,1,1]=>0
[4]=>1
[3,1]=>2
[2,2]=>0
[2,1,1]=>1
[1,1,1,1]=>0
[5]=>2
[4,1]=>3
[3,2]=>1
[3,1,1]=>1
[2,2,1]=>2
[2,1,1,1]=>1
[1,1,1,1,1]=>0
[6]=>5
[5,1]=>6
[4,2]=>3
[4,1,1]=>3
[3,3]=>2
[3,2,1]=>5
[3,1,1,1]=>2
[2,2,2]=>1
[2,2,1,1]=>1
[2,1,1,1,1]=>1
[1,1,1,1,1,1]=>0
[7]=>8
[6,1]=>9
[5,2]=>5
[5,1,1]=>5
[4,3]=>3
[4,2,1]=>7
[4,1,1,1]=>3
[3,3,1]=>6
[3,2,2]=>0
[3,2,1,1]=>3
[3,1,1,1,1]=>0
[2,2,2,1]=>5
[2,2,1,1,1]=>0
[2,1,1,1,1,1]=>1
[1,1,1,1,1,1,1]=>0
[8]=>14
[7,1]=>15
[6,2]=>10
[6,1,1]=>10
[5,3]=>7
[5,2,1]=>12
[5,1,1,1]=>7
[4,4]=>8
[4,3,1]=>10
[4,2,2]=>8
[4,2,1,1]=>12
[4,1,1,1,1]=>6
[3,3,2]=>4
[3,3,1,1]=>4
[3,2,2,1]=>7
[3,2,1,1,1]=>4
[3,1,1,1,1,1]=>1
[2,2,2,2]=>2
[2,2,2,1,1]=>6
[2,2,1,1,1,1]=>4
[2,1,1,1,1,1,1]=>2
[1,1,1,1,1,1,1,1]=>0
[9]=>21
[8,1]=>22
[7,2]=>16
[7,1,1]=>16
[6,3]=>12
[6,2,1]=>18
[6,1,1,1]=>12
[5,4]=>10
[5,3,1]=>15
[5,2,2]=>5
[5,2,1,1]=>10
[5,1,1,1,1]=>5
[4,4,1]=>16
[4,3,2]=>12
[4,3,1,1]=>12
[4,2,2,1]=>16
[4,2,1,1,1]=>12
[4,1,1,1,1,1]=>6
[3,3,3]=>8
[3,3,2,1]=>12
[3,3,1,1,1]=>13
[3,2,2,2]=>4
[3,2,2,1,1]=>6
[3,2,1,1,1,1]=>4
[3,1,1,1,1,1,1]=>6
[2,2,2,2,1]=>10
[2,2,2,1,1,1]=>4
[2,2,1,1,1,1,1]=>2
[2,1,1,1,1,1,1,1]=>2
[1,1,1,1,1,1,1,1,1]=>0
[10]=>32
[9,1]=>33
[8,2]=>26
[8,1,1]=>26
[7,3]=>21
[7,2,1]=>28
[7,1,1,1]=>21
[6,4]=>18
[6,3,1]=>24
[6,2,2]=>12
[6,2,1,1]=>18
[6,1,1,1,1]=>12
[5,5]=>22
[5,4,1]=>22
[5,3,2]=>18
[5,3,1,1]=>18
[5,2,2,1]=>17
[5,2,1,1,1]=>16
[5,1,1,1,1,1]=>12
[4,4,2]=>14
[4,4,1,1]=>14
[4,3,3]=>6
[4,3,2,1]=>18
[4,3,1,1,1]=>6
[4,2,2,2]=>10
[4,2,2,1,1]=>14
[4,2,1,1,1,1]=>10
[4,1,1,1,1,1,1]=>2
[3,3,3,1]=>20
[3,3,2,2]=>7
[3,3,2,1,1]=>6
[3,3,1,1,1,1]=>7
[3,2,2,2,1]=>12
[3,2,2,1,1,1]=>7
[3,2,1,1,1,1,1]=>4
[3,1,1,1,1,1,1,1]=>6
[2,2,2,2,2]=>6
[2,2,2,2,1,1]=>6
[2,2,2,1,1,1,1]=>10
[2,2,1,1,1,1,1,1]=>5
[2,1,1,1,1,1,1,1,1]=>2
[1,1,1,1,1,1,1,1,1,1]=>0
[11]=>45
[10,1]=>46
[9,2]=>38
[9,1,1]=>38
[8,3]=>32
[8,2,1]=>40
[8,1,1,1]=>32
[7,4]=>28
[7,3,1]=>35
[7,2,2]=>21
[7,2,1,1]=>28
[7,1,1,1,1]=>21
[6,5]=>26
[6,4,1]=>32
[6,3,2]=>20
[6,3,1,1]=>20
[6,2,2,1]=>26
[6,2,1,1,1]=>20
[6,1,1,1,1,1]=>14
[5,5,1]=>36
[5,4,2]=>22
[5,4,1,1]=>22
[5,3,3]=>11
[5,3,2,1]=>26
[5,3,1,1,1]=>11
[5,2,2,2]=>10
[5,2,2,1,1]=>12
[5,2,1,1,1,1]=>10
[5,1,1,1,1,1,1]=>6
[4,4,3]=>14
[4,4,2,1]=>28
[4,4,1,1,1]=>14
[4,3,3,1]=>20
[4,3,2,2]=>14
[4,3,2,1,1]=>26
[4,3,1,1,1,1]=>8
[4,2,2,2,1]=>24
[4,2,2,1,1,1]=>14
[4,2,1,1,1,1,1]=>14
[4,1,1,1,1,1,1,1]=>4
[3,3,3,2]=>12
[3,3,3,1,1]=>12
[3,3,2,2,1]=>11
[3,3,2,1,1,1]=>22
[3,3,1,1,1,1,1]=>5
[3,2,2,2,2]=>0
[3,2,2,2,1,1]=>8
[3,2,2,1,1,1,1]=>2
[3,2,1,1,1,1,1,1]=>10
[3,1,1,1,1,1,1,1,1]=>0
[2,2,2,2,2,1]=>20
[2,2,2,2,1,1,1]=>0
[2,2,2,1,1,1,1,1]=>4
[2,2,1,1,1,1,1,1,1]=>2
[2,1,1,1,1,1,1,1,1,1]=>2
[1,1,1,1,1,1,1,1,1,1,1]=>0
[12]=>65
[11,1]=>66
[10,2]=>57
[10,1,1]=>57
[9,3]=>50
[9,2,1]=>59
[9,1,1,1]=>50
[8,4]=>45
[8,3,1]=>53
[8,2,2]=>37
[8,2,1,1]=>45
[8,1,1,1,1]=>37
[7,5]=>42
[7,4,1]=>49
[7,3,2]=>35
[7,3,1,1]=>35
[7,2,2,1]=>42
[7,2,1,1,1]=>35
[7,1,1,1,1,1]=>28
[6,6]=>50
[6,5,1]=>47
[6,4,2]=>35
[6,4,1,1]=>35
[6,3,3]=>41
[6,3,2,1]=>47
[6,3,1,1,1]=>37
[6,2,2,2]=>26
[6,2,2,1,1]=>31
[6,2,1,1,1,1]=>26
[6,1,1,1,1,1,1]=>23
[5,5,2]=>37
[5,5,1,1]=>37
[5,4,3]=>29
[5,4,2,1]=>37
[5,4,1,1,1]=>25
[5,3,3,1]=>32
[5,3,2,2]=>18
[5,3,2,1,1]=>29
[5,3,1,1,1,1]=>14
[5,2,2,2,1]=>27
[5,2,2,1,1,1]=>16
[5,2,1,1,1,1,1]=>17
[5,1,1,1,1,1,1,1]=>12
[4,4,4]=>37
[4,4,3,1]=>35
[4,4,2,2]=>23
[4,4,2,1,1]=>37
[4,4,1,1,1,1]=>23
[4,3,3,2]=>19
[4,3,3,1,1]=>19
[4,3,2,2,1]=>39
[4,3,2,1,1,1]=>23
[4,3,1,1,1,1,1]=>15
[4,2,2,2,2]=>19
[4,2,2,2,1,1]=>19
[4,2,2,1,1,1,1]=>27
[4,2,1,1,1,1,1,1]=>11
[4,1,1,1,1,1,1,1,1]=>9
[3,3,3,3]=>26
[3,3,3,2,1]=>37
[3,3,3,1,1,1]=>26
[3,3,2,2,2]=>13
[3,3,2,2,1,1]=>12
[3,3,2,1,1,1,1]=>17
[3,3,1,1,1,1,1,1]=>8
[3,2,2,2,2,1]=>22
[3,2,2,2,1,1,1]=>17
[3,2,2,1,1,1,1,1]=>8
[3,2,1,1,1,1,1,1,1]=>15
[3,1,1,1,1,1,1,1,1,1]=>2
[2,2,2,2,2,2]=>12
[2,2,2,2,2,1,1]=>21
[2,2,2,2,1,1,1,1]=>12
[2,2,2,1,1,1,1,1,1]=>7
[2,2,1,1,1,1,1,1,1,1]=>2
[2,1,1,1,1,1,1,1,1,1,1]=>3
[1,1,1,1,1,1,1,1,1,1,1,1]=>0
                    

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Description

The number of characters of the symmetric group whose value on the partition is zero.
The maximal value for any given size is recorded in [2].

References

[1] Miller, A. R. Note on parity and the irreducible characters of the symmetric group arXiv:1708.03267
[2] Maximal number of zeros in a column of the character table of the symmetric group S_n. OEIS:A086642

Code

def statistic(la):
    s = SymmetricFunctions(ZZ).s()
    p = SymmetricFunctions(ZZ).p()
    P = Partitions(la.size())
    r = P.cardinality()
    res = [0]*r    
    for i, (mu, v) in enumerate(s(p(la))):
            res[i] = v
    return len([1 for e in res if is_even(e)])

Created

Aug 12, 2017 at 11:37 by Martin Rubey

Updated

Aug 12, 2017 at 13:44 by Martin Rubey