Identifier
-
Mp00119:
Dyck paths
—to 321-avoiding permutation (Krattenthaler)⟶
Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000240: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => 1
[1,0,1,0] => [1,2] => [1,2] => 2
[1,1,0,0] => [2,1] => [2,1] => 1
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => 3
[1,0,1,1,0,0] => [1,3,2] => [1,3,2] => 2
[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => 2
[1,1,0,1,0,0] => [2,3,1] => [3,1,2] => 3
[1,1,1,0,0,0] => [3,1,2] => [2,3,1] => 1
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => 3
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => 3
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,4,2,3] => 4
[1,0,1,1,1,0,0,0] => [1,4,2,3] => [1,3,4,2] => 2
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => 3
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [3,1,2,4] => 4
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,1,2,3] => 4
[1,1,0,1,1,0,0,0] => [2,4,1,3] => [3,1,4,2] => 3
[1,1,1,0,0,0,1,0] => [3,1,2,4] => [2,3,1,4] => 2
[1,1,1,0,0,1,0,0] => [3,1,4,2] => [2,4,1,3] => 3
[1,1,1,0,1,0,0,0] => [3,4,1,2] => [3,4,1,2] => 4
[1,1,1,1,0,0,0,0] => [4,1,2,3] => [2,3,4,1] => 1
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,4,3,5] => 4
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,5,3,4] => 5
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => [1,2,4,5,3] => 3
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,3,2,4,5] => 4
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,3,2,5,4] => 3
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,4,2,3,5] => 5
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,5,2,3,4] => 5
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => [1,4,2,5,3] => 4
[1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => [1,3,4,2,5] => 3
[1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => [1,3,5,2,4] => 4
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => [1,4,5,2,3] => 5
[1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => [1,3,4,5,2] => 2
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [2,1,3,4,5] => 4
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [2,1,3,5,4] => 3
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [2,1,4,3,5] => 3
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [2,1,5,3,4] => 4
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => [2,1,4,5,3] => 2
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [3,1,2,4,5] => 5
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [3,1,2,5,4] => 4
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [4,1,2,3,5] => 5
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => 5
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => [4,1,2,5,3] => 4
[1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => [3,1,4,2,5] => 4
[1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => [3,1,5,2,4] => 5
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => [4,1,5,2,3] => 5
[1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => [3,1,4,5,2] => 3
[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => [2,3,1,4,5] => 3
[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => [2,3,1,5,4] => 2
[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => [2,4,1,3,5] => 4
[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => [2,5,1,3,4] => 4
[1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => [2,4,1,5,3] => 3
[1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => [3,4,1,2,5] => 5
[1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => [3,5,1,2,4] => 5
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => [4,5,1,2,3] => 5
[1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => [3,4,1,5,2] => 4
[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => [2,3,4,1,5] => 2
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => [2,3,5,1,4] => 3
[1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => [2,4,5,1,3] => 4
[1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => [3,4,5,1,2] => 5
[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 5
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 5
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,6,4,5] => 6
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => [1,2,3,5,6,4] => 4
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 5
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => 4
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [1,2,5,3,4,6] => 6
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 6
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,3,5] => [1,2,5,3,6,4] => 5
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => [1,2,4,5,3,6] => 4
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => [1,2,4,6,3,5] => 5
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => 6
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => [1,2,4,5,6,3] => 3
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => 5
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => 4
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => 4
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [1,3,2,6,4,5] => 5
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => [1,3,2,5,6,4] => 3
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [1,4,2,3,5,6] => 6
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [1,4,2,3,6,5] => 5
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [1,5,2,3,4,6] => 6
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,6,2,3,4,5] => 6
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,2,5] => [1,5,2,3,6,4] => 5
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,2,4,6] => [1,4,2,5,3,6] => 5
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,2,6,4] => [1,4,2,6,3,5] => 6
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,2,4] => [1,5,2,6,3,4] => 6
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,2,4,5] => [1,4,2,5,6,3] => 4
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => [1,3,4,2,5,6] => 4
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => [1,3,4,2,6,5] => 3
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3,6] => [1,3,5,2,4,6] => 5
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => [1,3,6,2,4,5] => 5
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,2,6,3,5] => [1,3,5,2,6,4] => 4
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,2,3,6] => [1,4,5,2,3,6] => 6
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => [1,4,6,2,3,5] => 6
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => [1,5,6,2,3,4] => 6
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => [1,4,5,2,6,3] => 5
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Description
The number of indices that are not small excedances.
A small excedance is an index $i$ for which $\pi_i = i+1$.
A small excedance is an index $i$ for which $\pi_i = i+1$.
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.
Map
inverse
Description
Sends a permutation to its inverse.
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