Identifier
Values
[[1,2]] => [1,2] => [2,1] => [2,1] => 0
[[1],[2]] => [2,1] => [1,2] => [1,2] => 0
[[1,2,3]] => [1,2,3] => [3,2,1] => [3,2,1] => 1
[[1,3],[2]] => [2,1,3] => [3,1,2] => [3,1,2] => 0
[[1,2],[3]] => [3,1,2] => [2,1,3] => [2,1,3] => 0
[[1],[2],[3]] => [3,2,1] => [1,2,3] => [1,3,2] => 1
[[1,2,3,4]] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 2
[[1,3,4],[2]] => [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 2
[[1,2,4],[3]] => [3,1,2,4] => [4,2,1,3] => [4,2,1,3] => 1
[[1,2,3],[4]] => [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 1
[[1,3],[2,4]] => [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 0
[[1,2],[3,4]] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[[1,4],[2],[3]] => [3,2,1,4] => [4,1,2,3] => [4,1,3,2] => 0
[[1,3],[2],[4]] => [4,2,1,3] => [3,1,2,4] => [3,1,4,2] => 0
[[1,2],[3],[4]] => [4,3,1,2] => [2,1,3,4] => [2,1,4,3] => 0
[[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => [1,4,3,2] => 2
[[1,2,3,4,5]] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 3
[[1,3,4,5],[2]] => [2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,1,2] => 3
[[1,2,4,5],[3]] => [3,1,2,4,5] => [5,4,2,1,3] => [5,4,2,1,3] => 3
[[1,2,3,5],[4]] => [4,1,2,3,5] => [5,3,2,1,4] => [5,3,2,1,4] => 2
[[1,2,3,4],[5]] => [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => 2
[[1,3,5],[2,4]] => [2,4,1,3,5] => [5,3,1,4,2] => [5,3,1,4,2] => 2
[[1,2,5],[3,4]] => [3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => 1
[[1,3,4],[2,5]] => [2,5,1,3,4] => [4,3,1,5,2] => [4,3,1,5,2] => 2
[[1,2,4],[3,5]] => [3,5,1,2,4] => [4,2,1,5,3] => [4,2,1,5,3] => 1
[[1,2,3],[4,5]] => [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => 1
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,3,2] => 3
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [5,3,1,2,4] => [5,3,1,4,2] => 2
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [5,2,1,3,4] => [5,2,1,4,3] => 1
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [4,3,1,2,5] => [4,3,1,5,2] => 2
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [4,2,1,3,5] => [4,2,1,5,3] => 1
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,5,4] => 1
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [4,1,5,2,3] => [4,1,5,3,2] => 0
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [3,1,5,2,4] => [3,1,5,4,2] => 0
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [2,1,5,3,4] => [2,1,5,4,3] => 0
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [3,1,4,2,5] => [3,1,5,4,2] => 0
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [2,1,4,3,5] => [2,1,5,4,3] => 0
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [4,1,2,3,5] => [4,1,5,3,2] => 0
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [3,1,2,4,5] => [3,1,5,4,2] => 0
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [2,1,3,4,5] => [2,1,5,4,3] => 0
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,3,4,5] => [1,5,4,3,2] => 3
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Description
The number of inversions of the second entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$$
The number of inversions of the first entry is St000054The first entry of the permutation. and the number of inversions of the third entry is St001556The number of inversions of the third entry of a permutation.. The sequence of inversions of all the entries define the Lehmer code of a permutation.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
Map
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].