- FindStat

Identifier

St000618: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [1]=>1
[2]=>1
[1,1]=>1
[3]=>1
[2,1]=>0
[1,1,1]=>1
[4]=>1
[3,1]=>1
[2,2]=>2
[2,1,1]=>1
[1,1,1,1]=>1
[5]=>1
[4,1]=>0
[3,2]=>1
[3,1,1]=>2
[2,2,1]=>1
[2,1,1,1]=>0
[1,1,1,1,1]=>1
[6]=>1
[5,1]=>1
[4,2]=>3
[4,1,1]=>2
[3,3]=>3
[3,2,1]=>0
[3,1,1,1]=>2
[2,2,2]=>3
[2,2,1,1]=>3
[2,1,1,1,1]=>1
[1,1,1,1,1,1]=>1
[7]=>1
[6,1]=>0
[5,2]=>2
[5,1,1]=>3
[4,3]=>0
[4,2,1]=>1
[4,1,1,1]=>0
[3,3,1]=>3
[3,2,2]=>3
[3,2,1,1]=>1
[3,1,1,1,1]=>3
[2,2,2,1]=>0
[2,2,1,1,1]=>2
[2,1,1,1,1,1]=>0
[1,1,1,1,1,1,1]=>1
[8]=>1
[7,1]=>1
[6,2]=>4
[6,1,1]=>3
[5,3]=>4
[5,2,1]=>0
[5,1,1,1]=>3
[4,4]=>6
[4,3,1]=>2
[4,2,2]=>8
[4,2,1,1]=>6
[4,1,1,1,1]=>3
[3,3,2]=>6
[3,3,1,1]=>8
[3,2,2,1]=>2
[3,2,1,1,1]=>0
[3,1,1,1,1,1]=>3
[2,2,2,2]=>6
[2,2,2,1,1]=>4
[2,2,1,1,1,1]=>4
[2,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,1,1]=>1
[9]=>1
[8,1]=>0
[7,2]=>3
[7,1,1]=>4
[6,3]=>0
[6,2,1]=>1
[6,1,1,1]=>0
[5,4]=>2
[5,3,1]=>6
[5,2,2]=>8
[5,2,1,1]=>3
[5,1,1,1,1]=>6
[4,4,1]=>4
[4,3,2]=>0
[4,3,1,1]=>0
[4,2,2,1]=>0
[4,2,1,1,1]=>3
[4,1,1,1,1,1]=>0
[3,3,3]=>6
[3,3,2,1]=>0
[3,3,1,1,1]=>8
[3,2,2,2]=>4
[3,2,2,1,1]=>6
[3,2,1,1,1,1]=>1
[3,1,1,1,1,1,1]=>4
[2,2,2,2,1]=>2
[2,2,2,1,1,1]=>0
[2,2,1,1,1,1,1]=>3
[2,1,1,1,1,1,1,1]=>0
[1,1,1,1,1,1,1,1,1]=>1
[10]=>1
[9,1]=>1
[8,2]=>5
[8,1,1]=>4
[7,3]=>5
[7,2,1]=>0
[7,1,1,1]=>4
[6,4]=>10
[6,3,1]=>5
[6,2,2]=>15
[6,2,1,1]=>10
[6,1,1,1,1]=>6
[5,5]=>10
[5,4,1]=>0
[5,3,2]=>10
[5,3,1,1]=>15
[5,2,2,1]=>5
[5,2,1,1,1]=>0
[5,1,1,1,1,1]=>6
[4,4,2]=>20
[4,4,1,1]=>20
[4,3,3]=>10
[4,3,2,1]=>0
[4,3,1,1,1]=>5
[4,2,2,2]=>20
[4,2,2,1,1]=>15
[4,2,1,1,1,1]=>10
[4,1,1,1,1,1,1]=>4
[3,3,3,1]=>10
[3,3,2,2]=>20
[3,3,2,1,1]=>10
[3,3,1,1,1,1]=>15
[3,2,2,2,1]=>0
[3,2,2,1,1,1]=>5
[3,2,1,1,1,1,1]=>0
[3,1,1,1,1,1,1,1]=>4
[2,2,2,2,2]=>10
[2,2,2,2,1,1]=>10
[2,2,2,1,1,1,1]=>5
[2,2,1,1,1,1,1,1]=>5
[2,1,1,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,1,1,1,1]=>1
[11]=>1
[10,1]=>0
[9,2]=>4
[9,1,1]=>5
[8,3]=>0
[8,2,1]=>1
[8,1,1,1]=>0
[7,4]=>5
[7,3,1]=>10
[7,2,2]=>15
[7,2,1,1]=>6
[7,1,1,1,1]=>10
[6,5]=>0
[6,4,1]=>5
[6,3,2]=>0
[6,3,1,1]=>0
[6,2,2,1]=>0
[6,2,1,1,1]=>4
[6,1,1,1,1,1]=>0
[5,5,1]=>10
[5,4,2]=>10
[5,4,1,1]=>5
[5,3,3]=>20
[5,3,2,1]=>0
[5,3,1,1,1]=>20
[5,2,2,2]=>15
[5,2,2,1,1]=>20
[5,2,1,1,1,1]=>4
[5,1,1,1,1,1,1]=>10
[4,4,3]=>10
[4,4,2,1]=>0
[4,4,1,1,1]=>15
[4,3,3,1]=>0
[4,3,2,2]=>0
[4,3,2,1,1]=>0
[4,3,1,1,1,1]=>0
[4,2,2,2,1]=>5
[4,2,2,1,1,1]=>0
[4,2,1,1,1,1,1]=>6
[4,1,1,1,1,1,1,1]=>0
[3,3,3,2]=>10
[3,3,3,1,1]=>20
[3,3,2,2,1]=>10
[3,3,2,1,1,1]=>0
[3,3,1,1,1,1,1]=>15
[3,2,2,2,2]=>10
[3,2,2,2,1,1]=>5
[3,2,2,1,1,1,1]=>10
[3,2,1,1,1,1,1,1]=>1
[3,1,1,1,1,1,1,1,1]=>5
[2,2,2,2,2,1]=>0
[2,2,2,2,1,1,1]=>5
[2,2,2,1,1,1,1,1]=>0
[2,2,1,1,1,1,1,1,1]=>4
[2,1,1,1,1,1,1,1,1,1]=>0
[1,1,1,1,1,1,1,1,1,1,1]=>1
[12]=>1
[11,1]=>1
[10,2]=>6
[10,1,1]=>5
[9,3]=>6
[9,2,1]=>0
[9,1,1,1]=>5
[8,4]=>15
[8,3,1]=>9
[8,2,2]=>24
[8,2,1,1]=>15
[8,1,1,1,1]=>10
[7,5]=>15
[7,4,1]=>0
[7,3,2]=>15
[7,3,1,1]=>24
[7,2,2,1]=>9
[7,2,1,1,1]=>0
[7,1,1,1,1,1]=>10
[6,6]=>20
[6,5,1]=>5
[6,4,2]=>45
[6,4,1,1]=>40
[6,3,3]=>30
[6,3,2,1]=>0
[6,3,1,1,1]=>16
[6,2,2,2]=>45
[6,2,2,1,1]=>36
[6,2,1,1,1,1]=>20
[6,1,1,1,1,1,1]=>10
[5,5,2]=>40
[5,5,1,1]=>45
[5,4,3]=>0
[5,4,2,1]=>5
[5,4,1,1,1]=>0
[5,3,3,1]=>30
[5,3,2,2]=>45
[5,3,2,1,1]=>20
[5,3,1,1,1,1]=>36
[5,2,2,2,1]=>0
[5,2,2,1,1,1]=>16
[5,2,1,1,1,1,1]=>0
[5,1,1,1,1,1,1,1]=>10
[4,4,4]=>30
[4,4,3,1]=>30
[4,4,2,2]=>80
[4,4,2,1,1]=>45
[4,4,1,1,1,1]=>45
[4,3,3,2]=>30
[4,3,3,1,1]=>30
[4,3,2,2,1]=>5
[4,3,2,1,1,1]=>0
[4,3,1,1,1,1,1]=>9
[4,2,2,2,2]=>45
[4,2,2,2,1,1]=>40
[4,2,2,1,1,1,1]=>24
[4,2,1,1,1,1,1,1]=>15
[4,1,1,1,1,1,1,1,1]=>5
[3,3,3,3]=>30
[3,3,3,2,1]=>0
[3,3,3,1,1,1]=>30
[3,3,2,2,2]=>40
[3,3,2,2,1,1]=>45
[3,3,2,1,1,1,1]=>15
[3,3,1,1,1,1,1,1]=>24
[3,2,2,2,2,1]=>5
[3,2,2,2,1,1,1]=>0
[3,2,2,1,1,1,1,1]=>9
[3,2,1,1,1,1,1,1,1]=>0
[3,1,1,1,1,1,1,1,1,1]=>5
[2,2,2,2,2,2]=>20
[2,2,2,2,2,1,1]=>15
[2,2,2,2,1,1,1,1]=>15
[2,2,2,1,1,1,1,1,1]=>6
[2,2,1,1,1,1,1,1,1,1]=>6
[2,1,1,1,1,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,1,1,1,1,1,1]=>1
                    

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Description

The number of self-evacuating tableaux of given shape.
This is the same as the number of standard domino tableaux of the given shape.

References

[1] Stembridge, J. R. Canonical bases and self-evacuating tableaux MathSciNet:1387685

Code

def statistic_alt(la):
    chi = SymmetricGroupRepresentation(la)
    r = abs(ZZ(chi.to_character()(Permutation(range(la.size(),0,-1)))))
    assert r==statistic(la)
    return r

def statistic(la):
    n = la.size()
    la = la + [0]*(n-len(la))

    E_la = sorted([la[j] + n-j-1 for j in range(n) if is_even(la[j] + n-j-1)], reverse=True)
    rE = len(E_la)
    la_e = [ZZ(E_la[i]/2-rE+i+1) for i in range(rE)]
    
    O_la = sorted([la[j] + n-j-2 for j in range(n) if is_even(la[j] +n-j-2)], reverse=True)
    rO = len(O_la)
    la_o = [ZZ(O_la[i]/2-rO+i+1) for i in range(rO)]

    if abs(rE - rO)<=1:
        return (binomial(n//2, sum(la_e))*
                StandardTableaux(la_e).cardinality()*
                StandardTableaux(la_o).cardinality())
    else:
        return 0

Created

Sep 26, 2016 at 23:00 by Martin Rubey

Updated

Sep 26, 2016 at 23:00 by Martin Rubey