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St001081: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 1
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 3
[1,4,2,3] => 3
[1,4,3,2] => 2
[2,1,3,4] => 1
[2,1,4,3] => 4
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 4
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 4
[1,2,3,4,5] => 1
[1,2,3,5,4] => 2
[1,2,4,3,5] => 2
[1,2,4,5,3] => 3
[1,2,5,3,4] => 3
[1,2,5,4,3] => 2
[1,3,2,4,5] => 2
[1,3,2,5,4] => 24
[1,3,4,2,5] => 3
[1,3,4,5,2] => 4
[1,3,5,2,4] => 4
[1,3,5,4,2] => 3
[1,4,2,3,5] => 3
[1,4,2,5,3] => 4
[1,4,3,2,5] => 2
[1,4,3,5,2] => 3
[1,4,5,2,3] => 24
[1,4,5,3,2] => 4
Description
The number of minimal length factorizations of a permutation into star transpositions. For a permutation πSn a minimal length factorization into star transpositions is a factorization of the form π=τi1τik,2i1,,ikn, where τa=(1,a) for 2an and k is minimal. [1, lem.2.1] shows that the minimal length of such a factorization is n+ma1, where m is the number of non-trival cycles not containing the element 1, and a is the number of fixed points different from 1, see [[St001077]]. [2, cor.2] shows that the number of such minimal factorizations is (n+m2(k+1))!(nk)!1m, where 1,,m is the cycle type of π and k is the number of fixed point different from 1.