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Matching statistic: St001639
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Mapping
St001639: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 3
[1,3,2,4] => 2
[1,3,4,2] => 3
[1,4,2,3] => 3
[1,4,3,2] => 1
[2,1,3,4] => 3
[2,1,4,3] => 2
[2,3,1,4] => 3
[2,3,4,1] => 1
[2,4,1,3] => 4
[2,4,3,1] => 3
[3,1,2,4] => 3
[3,1,4,2] => 4
[3,2,1,4] => 1
[3,2,4,1] => 3
[3,4,1,2] => 2
[3,4,2,1] => 3
[4,1,2,3] => 1
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 2
[4,3,1,2] => 3
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 6
[1,2,4,3,5] => 5
[1,2,4,5,3] => 7
[1,2,5,3,4] => 7
[1,2,5,4,3] => 3
[1,3,2,4,5] => 5
[1,3,2,5,4] => 6
[1,3,4,2,5] => 6
[1,3,4,5,2] => 5
[1,3,5,2,4] => 9
[1,3,5,4,2] => 7
[1,4,2,3,5] => 6
[1,4,2,5,3] => 9
[1,4,3,2,5] => 2
[1,4,3,5,2] => 7
[1,4,5,2,3] => 4
Description
The number of alternating subsets such that applying the permutation does not yield an alternating subset. A subset of $[n]=\{1,\dots,n\}$ is alternating if any two successive elements have different parity. This statistic records for each permutation $\pi\in\mathfrak S_n$ the number of alternating subsets $S\subseteq [n]$ such that $\pi(S)$ is not alternating. Note that the number of alternating subsets of $[n]$ is $F(n+3)-1$, where $F(n)$ is the $n$-th Fibonacci number.