- FindStat

Identifier

St001364: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [1]=>1
[2]=>1
[1,1]=>1
[3]=>0
[2,1]=>1
[1,1,1]=>3
[4]=>1
[3,1]=>0
[2,2]=>1
[2,1,1]=>1
[1,1,1,1]=>9
[5]=>1
[4,1]=>1
[3,2]=>0
[3,1,1]=>0
[2,2,1]=>1
[2,1,1,1]=>3
[1,1,1,1,1]=>21
[6]=>0
[5,1]=>1
[4,2]=>1
[4,1,1]=>1
[3,3]=>0
[3,2,1]=>0
[3,1,1,1]=>0
[2,2,2]=>9
[2,2,1,1]=>1
[2,1,1,1,1]=>9
[1,1,1,1,1,1]=>81
[7]=>1
[6,1]=>0
[5,2]=>1
[5,1,1]=>1
[4,3]=>0
[4,2,1]=>1
[4,1,1,1]=>3
[3,3,1]=>0
[3,2,2]=>0
[3,2,1,1]=>0
[3,1,1,1,1]=>0
[2,2,2,1]=>9
[2,2,1,1,1]=>3
[2,1,1,1,1,1]=>21
[1,1,1,1,1,1,1]=>351
[8]=>1
[7,1]=>1
[6,2]=>0
[6,1,1]=>0
[5,3]=>0
[5,2,1]=>1
[5,1,1,1]=>3
[4,4]=>1
[4,3,1]=>0
[4,2,2]=>1
[4,2,1,1]=>1
[4,1,1,1,1]=>9
[3,3,2]=>0
[3,3,1,1]=>0
[3,2,2,1]=>0
[3,2,1,1,1]=>0
[3,1,1,1,1,1]=>0
[2,2,2,2]=>33
[2,2,2,1,1]=>9
[2,2,1,1,1,1]=>9
[2,1,1,1,1,1,1]=>81
[1,1,1,1,1,1,1,1]=>1233
[9]=>0
[8,1]=>1
[7,2]=>1
[7,1,1]=>1
[6,3]=>0
[6,2,1]=>0
[6,1,1,1]=>0
[5,4]=>1
[5,3,1]=>0
[5,2,2]=>1
[5,2,1,1]=>1
[5,1,1,1,1]=>9
[4,4,1]=>1
[4,3,2]=>0
[4,3,1,1]=>0
[4,2,2,1]=>1
[4,2,1,1,1]=>3
[4,1,1,1,1,1]=>21
[3,3,3]=>18
[3,3,2,1]=>0
[3,3,1,1,1]=>0
[3,2,2,2]=>0
[3,2,2,1,1]=>0
[3,2,1,1,1,1]=>0
[3,1,1,1,1,1,1]=>0
[2,2,2,2,1]=>33
[2,2,2,1,1,1]=>27
[2,2,1,1,1,1,1]=>21
[2,1,1,1,1,1,1,1]=>351
[1,1,1,1,1,1,1,1,1]=>5769
                    

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Description

The number of permutations whose cube equals a fixed permutation of given cycle type.
For example, the permutation $\pi=412365$ has cycle type $(4,2)$ and $234165$ is the unique permutation whose cube is $\pi$.

Code

@cached_function
def statistic_dict(n, k):
    d = {}
    for pi in Permutations(n):
        sigma = pi^k
        d[sigma] = d.get(sigma, 0) + 1
    return d

def statistic(la):
    n = la.size()
    d = statistic_dict(n, 3)
    sigma = standard_permutation(la)
    return d.get(sigma, 0)

Created

Mar 15, 2019 at 20:50 by Martin Rubey

Updated

Mar 15, 2019 at 20:50 by Martin Rubey