- FindStat

Identifier

St001710: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        []=>1
[1]=>1
[2]=>2
[1,1]=>2
[3]=>1
[2,1]=>4
[1,1,1]=>4
[4]=>2
[3,1]=>1
[2,2]=>10
[2,1,1]=>10
[1,1,1,1]=>10
[5]=>1
[4,1]=>2
[3,2]=>2
[3,1,1]=>2
[2,2,1]=>26
[2,1,1,1]=>26
[1,1,1,1,1]=>26
[6]=>4
[5,1]=>1
[4,2]=>4
[4,1,1]=>4
[3,3]=>4
[3,2,1]=>4
[3,1,1,1]=>4
[2,2,2]=>76
[2,2,1,1]=>76
[2,1,1,1,1]=>76
[1,1,1,1,1,1]=>76
[7]=>1
[6,1]=>4
[5,2]=>2
[5,1,1]=>2
[4,3]=>2
[4,2,1]=>8
[4,1,1,1]=>8
[3,3,1]=>4
[3,2,2]=>10
[3,2,1,1]=>10
[3,1,1,1,1]=>10
[2,2,2,1]=>232
[2,2,1,1,1]=>232
[2,1,1,1,1,1]=>232
[1,1,1,1,1,1,1]=>232
[8]=>4
[7,1]=>1
[6,2]=>8
[6,1,1]=>8
[5,3]=>1
[5,2,1]=>4
[5,1,1,1]=>4
[4,4]=>12
[4,3,1]=>2
[4,2,2]=>20
[4,2,1,1]=>20
[4,1,1,1,1]=>20
[3,3,2]=>8
[3,3,1,1]=>8
[3,2,2,1]=>26
[3,2,1,1,1]=>26
[3,1,1,1,1,1]=>26
[2,2,2,2]=>764
[2,2,2,1,1]=>764
[2,2,1,1,1,1]=>764
[2,1,1,1,1,1,1]=>764
[1,1,1,1,1,1,1,1]=>764
                    

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation.
Let $\alpha$ be any permutation of cycle type $\lambda$. This statistic is the number of permutations $\pi$ such that
$$ \alpha\pi\alpha^{-1} = \pi^{-1}.$$

References

[1] Homolya, S., Szigeti, Jenő Solving equations in the symmetric group arXiv:2104.03593

Code

def statistic(la):
    total = 0
    cycles = []
    for p in la:
        cycles.append(tuple(range(total+1, total+p+1)))
        total += p
    a = Permutation(cycles)
    return sum(1 for pi in Permutations(len(a)) if pi*a*pi == a)

Created

Apr 09, 2021 at 11:05 by Martin Rubey

Updated

Apr 09, 2021 at 11:05 by Martin Rubey