Identifier
-
Mp00119:
Dyck paths
—to 321-avoiding permutation (Krattenthaler)⟶
Permutations
St000021: Permutations ⟶ ℤ (values match St000325The width of the tree associated to a permutation., St000470The number of runs in a permutation.)
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] => 1
[1,1,1,0,0,0] => [3,1,2] => 1
[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] => 1
[1,0,1,1,1,0,0,0] => [1,4,2,3] => 1
[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] => 1
[1,1,0,1,0,1,0,0] => [2,3,4,1] => 1
[1,1,0,1,1,0,0,0] => [2,4,1,3] => 1
[1,1,1,0,0,0,1,0] => [3,1,2,4] => 1
[1,1,1,0,0,1,0,0] => [3,1,4,2] => 2
[1,1,1,0,1,0,0,0] => [3,4,1,2] => 1
[1,1,1,1,0,0,0,0] => [4,1,2,3] => 1
[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] => 1
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => 1
[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] => 1
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 1
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => 1
[1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => 1
[1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => 2
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => 1
[1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => 1
[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] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => 2
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 2
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 1
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 1
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => 1
[1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => 1
[1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => 2
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => 1
[1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => 1
[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => 1
[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => 2
[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => 2
[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => 2
[1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => 2
[1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => 1
[1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => 2
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => 1
[1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => 1
[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => 1
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => 2
[1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => 2
[1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => 1
[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => 1
[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] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => 1
[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] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,3,5] => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => 1
[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] => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => 2
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,2,5] => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,2,4,6] => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,2,6,4] => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,2,4] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,2,4,5] => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3,6] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,2,6,3,5] => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,2,3,6] => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => 1
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Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
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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