Identifier
-
Mp00119:
Dyck paths
—to 321-avoiding permutation (Krattenthaler)⟶
Permutations
St001579: Permutations ⟶ ℤ
Values
[1,0] => [1] => 0
[1,0,1,0] => [1,2] => 0
[1,1,0,0] => [2,1] => 1
[1,0,1,0,1,0] => [1,2,3] => 0
[1,0,1,1,0,0] => [1,3,2] => 1
[1,1,0,0,1,0] => [2,1,3] => 1
[1,1,0,1,0,0] => [2,3,1] => 2
[1,1,1,0,0,0] => [3,1,2] => 2
[1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0] => [1,3,4,2] => 2
[1,0,1,1,1,0,0,0] => [1,4,2,3] => 2
[1,1,0,0,1,0,1,0] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0] => [2,3,1,4] => 2
[1,1,0,1,0,1,0,0] => [2,3,4,1] => 3
[1,1,0,1,1,0,0,0] => [2,4,1,3] => 3
[1,1,1,0,0,0,1,0] => [3,1,2,4] => 2
[1,1,1,0,0,1,0,0] => [3,1,4,2] => 3
[1,1,1,0,1,0,0,0] => [3,4,1,2] => 4
[1,1,1,1,0,0,0,0] => [4,1,2,3] => 3
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 2
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => 2
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => 2
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 3
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => 3
[1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => 2
[1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => 3
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => 4
[1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => 3
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 3
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => 3
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 3
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 3
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 4
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => 4
[1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => 3
[1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => 4
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => 5
[1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => 4
[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => 2
[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => 3
[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => 3
[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => 4
[1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => 4
[1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => 4
[1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => 5
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => 6
[1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => 5
[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => 3
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => 4
[1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => 5
[1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => 6
[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => 2
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 3
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,3,5] => 3
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => 3
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => 4
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => 3
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => 3
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => 3
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 4
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,2,5] => 4
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,2,4,6] => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,2,6,4] => 4
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,2,4] => 5
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,2,4,5] => 4
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => 3
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3,6] => 3
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => 4
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,2,6,3,5] => 4
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,2,3,6] => 4
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => 5
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => 6
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => 5
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Description
The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation.
This is for a permutation $\sigma$ of length $n$ and the set $T = \{ (1,2), \dots, (n-1,n), (1,n) \}$ given by
$$\min\{ k \mid \sigma = t_1\dots t_k \text{ for } t_i \in T \text{ such that } t_1\dots t_j \text{ has more cyclic descents than } t_1\dots t_{j-1} \text{ for all } j\}.$$
This is for a permutation $\sigma$ of length $n$ and the set $T = \{ (1,2), \dots, (n-1,n), (1,n) \}$ given by
$$\min\{ k \mid \sigma = t_1\dots t_k \text{ for } t_i \in T \text{ such that } t_1\dots t_j \text{ has more cyclic descents than } t_1\dots t_{j-1} \text{ for all } j\}.$$
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
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