Identifier
-
Mp00119:
Dyck paths
—to 321-avoiding permutation (Krattenthaler)⟶
Permutations
St000039: Permutations ⟶ ℤ
Values
[1,0] => [1] => 0
[1,0,1,0] => [1,2] => 0
[1,1,0,0] => [2,1] => 0
[1,0,1,0,1,0] => [1,2,3] => 0
[1,0,1,1,0,0] => [1,3,2] => 0
[1,1,0,0,1,0] => [2,1,3] => 0
[1,1,0,1,0,0] => [2,3,1] => 1
[1,1,1,0,0,0] => [3,1,2] => 0
[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] => 0
[1,0,1,1,0,0,1,0] => [1,3,2,4] => 0
[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] => 0
[1,1,0,0,1,0,1,0] => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0] => [2,1,4,3] => 0
[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] => 2
[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] => 0
[1,1,1,0,0,1,0,0] => [3,1,4,2] => 1
[1,1,1,0,1,0,0,0] => [3,4,1,2] => 2
[1,1,1,1,0,0,0,0] => [4,1,2,3] => 0
[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] => 0
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 0
[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] => 0
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 0
[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] => 2
[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] => 0
[1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => 1
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => 2
[1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 0
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 1
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => 0
[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] => 1
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 2
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 3
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => 2
[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] => 3
[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] => 0
[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => 0
[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => 1
[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] => 1
[1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => 2
[1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => 3
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => 4
[1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => 2
[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => 0
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => 1
[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] => 3
[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => 0
[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] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 0
[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] => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 0
[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] => 2
[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] => 0
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 0
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => 1
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => 0
[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] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 3
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,2,5] => 2
[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] => 3
[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] => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3,6] => 1
[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] => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,2,3,6] => 2
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => 3
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => 4
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => 2
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Description
The number of crossings of a permutation.
A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$.
Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$.
Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
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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