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Identifier

St001731: Permutations ⟶ ℤ

Values

[1,2] => 0

[2,1] => 0

[1,2,3] => 0

[1,3,2] => 0

[2,1,3] => 0

[2,3,1] => 1

[3,1,2] => 1

[3,2,1] => 0

[1,2,3,4] => 0

[1,2,4,3] => 0

[1,3,2,4] => 0

[1,3,4,2] => 1

[1,4,2,3] => 1

[1,4,3,2] => 0

[2,1,3,4] => 0

[2,1,4,3] => 1

[2,3,1,4] => 1

[2,3,4,1] => 6

[2,4,1,3] => 6

[2,4,3,1] => 1

[3,1,2,4] => 1

[3,1,4,2] => 6

[3,2,1,4] => 0

[3,2,4,1] => 1

[3,4,1,2] => 1

[3,4,2,1] => 6

[4,1,2,3] => 6

[4,1,3,2] => 1

[4,2,1,3] => 1

[4,2,3,1] => 0

[4,3,1,2] => 6

[4,3,2,1] => 1

[1,2,3,4,5] => 0

[1,2,3,5,4] => 0

[1,2,4,3,5] => 0

[1,2,4,5,3] => 1

[1,2,5,3,4] => 1

[1,2,5,4,3] => 0

[1,3,2,4,5] => 0

[1,3,2,5,4] => 1

[1,3,4,2,5] => 1

[1,3,4,5,2] => 6

[1,3,5,2,4] => 6

[1,3,5,4,2] => 1

[1,4,2,3,5] => 1

[1,4,2,5,3] => 6

[1,4,3,2,5] => 0

[1,4,3,5,2] => 1

[1,4,5,2,3] => 1

[1,4,5,3,2] => 6

[1,5,2,3,4] => 6

[1,5,2,4,3] => 1

[1,5,3,2,4] => 1

[1,5,3,4,2] => 0

[1,5,4,2,3] => 6

[1,5,4,3,2] => 1

[2,1,3,4,5] => 0

[2,1,3,5,4] => 1

[2,1,4,3,5] => 1

[2,1,4,5,3] => 4

[2,1,5,3,4] => 4

[2,1,5,4,3] => 1

[2,3,1,4,5] => 1

[2,3,1,5,4] => 4

[2,3,4,1,5] => 6

[2,3,4,5,1] => 20

[2,3,5,1,4] => 20

[2,3,5,4,1] => 6

[2,4,1,3,5] => 6

[2,4,1,5,3] => 20

[2,4,3,1,5] => 1

[2,4,3,5,1] => 6

[2,4,5,1,3] => 4

[2,4,5,3,1] => 20

[2,5,1,3,4] => 20

[2,5,1,4,3] => 6

[2,5,3,1,4] => 6

[2,5,3,4,1] => 1

[2,5,4,1,3] => 20

[2,5,4,3,1] => 4

[3,1,2,4,5] => 1

[3,1,2,5,4] => 4

[3,1,4,2,5] => 6

[3,1,4,5,2] => 20

[3,1,5,2,4] => 20

[3,1,5,4,2] => 6

[3,2,1,4,5] => 0

[3,2,1,5,4] => 1

[3,2,4,1,5] => 1

[3,2,4,5,1] => 6

[3,2,5,1,4] => 6

[3,2,5,4,1] => 1

[3,4,1,2,5] => 1

[3,4,1,5,2] => 4

[3,4,2,1,5] => 6

[3,4,2,5,1] => 20

[3,4,5,1,2] => 20

[3,4,5,2,1] => 4

[3,5,1,2,4] => 4

[3,5,1,4,2] => 1

[3,5,2,1,4] => 20

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Description

The factorization defect of a permutation.
The factorization poset of a permutation $\sigma$ is the principal order ideal generated by $\sigma$ in the absolute order. In particular, the maximal chains in the factorization poset of $\sigma$ are in bijection with the reduced factorizations of $\sigma$ into transpositions. The factorization defect of $\sigma$ is the number of rank-2 elements in its factorization poset.
This is the statistic "d" defined in [1, Section 6.1], where it was called the minimal size of a feedback arc set in the cycle graph. The fact that this equals the number of rank-2 elements in the factorization poset follows from [1, Proposition 6.3] together with the shellability of the factorization poset.

References

[1] Mühle, H., Ripoll, V. Connectivity Properties of Factorization Posets in Generated Groups DOI:10.1007/s11083-019-09496-1

Code

def narayana_number(n,k):
    return 1/n*binomial(n,k)*binomial(n,k-1)

def statistic(pi):
    c = tuple(pi.cycle_type())
    return sum([narayana_number(c[T[0]],2)*narayana_number(c[T[1]],2) for T in Subsets(range(len(c)),2)]) + sum([narayana_number(t,3) for t in c])

Created

Jul 07, 2021 at 10:23 by Henri Mühle

Updated

Nov 18, 2024 at 19:29 by Nupur Jain