Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St001731
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Mapping
St001731: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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.