Identifier
Values
[2] => [[1,2]] => [1,2] => [1,2] => 1
[1,1] => [[1],[2]] => [2,1] => [2,1] => 1
[3] => [[1,2,3]] => [1,2,3] => [1,3,2] => 1
[2,1] => [[1,3],[2]] => [2,1,3] => [2,1,3] => 1
[1,1,1] => [[1],[2],[3]] => [3,2,1] => [3,2,1] => 1
[4] => [[1,2,3,4]] => [1,2,3,4] => [1,4,3,2] => 1
[3,1] => [[1,3,4],[2]] => [2,1,3,4] => [2,1,4,3] => 1
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => [3,4,1,2] => 1
[2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [4,3,2,1] => 1
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [1,5,4,3,2] => 1
[4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => [2,1,5,4,3] => 1
[3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => [3,5,1,4,2] => 2
[3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,2,1,5,4] => 1
[2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [4,2,5,1,3] => 2
[2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [1,6,5,4,3,2] => 1
[5,1] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [2,1,6,5,4,3] => 1
[4,2] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [3,6,1,5,4,2] => 2
[4,1,1] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [3,2,1,6,5,4] => 1
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [4,6,5,1,3,2] => 2
[3,2,1] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [4,2,6,1,5,3] => 3
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [4,3,2,1,6,5] => 1
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [5,6,3,4,1,2] => 2
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => [5,3,2,6,1,4] => 2
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => 1
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2] => [7,8,5,6,3,4,1,2] => 3
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Description
The number of cyclic valleys and cyclic peaks of a permutation.
This is given by the number of indices $i$ such that $\pi_{i-1} > \pi_i < \pi_{i+1}$ with indices considered cyclically. Equivalently, this is the number of indices $i$ such that $\pi_{i-1} < \pi_i > \pi_{i+1}$ with indices considered cyclically.
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
reading tableau
Description
Return the RSK recording tableau of the reading word of the (standard) tableau $T$ labeled down (in English convention) each column to the shape of a partition.
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].