Identifier
-
Mp00119:
Dyck paths
—to 321-avoiding permutation (Krattenthaler)⟶
Permutations
Mp00069: Permutations —complement⟶ Permutations
St001663: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => 0
[1,0,1,0] => [1,2] => [2,1] => 0
[1,1,0,0] => [2,1] => [1,2] => 0
[1,0,1,0,1,0] => [1,2,3] => [3,2,1] => 0
[1,0,1,1,0,0] => [1,3,2] => [3,1,2] => 0
[1,1,0,0,1,0] => [2,1,3] => [2,3,1] => 0
[1,1,0,1,0,0] => [2,3,1] => [2,1,3] => 0
[1,1,1,0,0,0] => [3,1,2] => [1,3,2] => 1
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [4,3,1,2] => 0
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [4,2,3,1] => 0
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [4,2,1,3] => 0
[1,0,1,1,1,0,0,0] => [1,4,2,3] => [4,1,3,2] => 1
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [3,4,2,1] => 0
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [3,4,1,2] => 0
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [3,2,4,1] => 0
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [3,2,1,4] => 0
[1,1,0,1,1,0,0,0] => [2,4,1,3] => [3,1,4,2] => 0
[1,1,1,0,0,0,1,0] => [3,1,2,4] => [2,4,3,1] => 1
[1,1,1,0,0,1,0,0] => [3,1,4,2] => [2,4,1,3] => 0
[1,1,1,0,1,0,0,0] => [3,4,1,2] => [2,1,4,3] => 0
[1,1,1,1,0,0,0,0] => [4,1,2,3] => [1,4,3,2] => 0
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [5,4,3,1,2] => 0
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [5,4,2,3,1] => 0
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [5,4,2,1,3] => 0
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => [5,4,1,3,2] => 1
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [5,3,4,2,1] => 0
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [5,3,4,1,2] => 0
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [5,3,2,4,1] => 0
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [5,3,2,1,4] => 0
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => [5,3,1,4,2] => 0
[1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => [5,2,4,3,1] => 1
[1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => [5,2,4,1,3] => 0
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => [5,2,1,4,3] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => [5,1,4,3,2] => 0
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [4,5,3,2,1] => 0
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [4,5,3,1,2] => 0
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [4,5,2,3,1] => 0
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [4,5,2,1,3] => 0
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => [4,5,1,3,2] => 1
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [4,3,5,2,1] => 0
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [4,3,5,1,2] => 0
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [4,3,2,5,1] => 0
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [4,3,2,1,5] => 0
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => [4,3,1,5,2] => 0
[1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => [4,2,5,3,1] => 0
[1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => [4,2,5,1,3] => 0
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => [4,2,1,5,3] => 0
[1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => [4,1,5,3,2] => 0
[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => [3,5,4,2,1] => 1
[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => [3,5,4,1,2] => 1
[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => [3,5,2,4,1] => 0
[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => [3,5,2,1,4] => 0
[1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => [3,5,1,4,2] => 0
[1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => [3,2,5,4,1] => 0
[1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => [3,2,5,1,4] => 0
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => [3,2,1,5,4] => 0
[1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => [3,1,5,4,2] => 0
[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => [2,5,4,3,1] => 0
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => [2,5,4,1,3] => 0
[1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => [2,5,1,4,3] => 0
[1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => [2,1,5,4,3] => 0
[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => [1,5,4,3,2] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [6,5,4,3,1,2] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [6,5,4,2,3,1] => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [6,5,4,2,1,3] => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => [6,5,4,1,3,2] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [6,5,3,4,1,2] => 0
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [6,5,3,2,4,1] => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [6,5,3,2,1,4] => 0
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,3,5] => [6,5,3,1,4,2] => 0
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => [6,5,2,4,3,1] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => [6,5,2,4,1,3] => 0
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => [6,5,2,1,4,3] => 0
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => [6,5,1,4,3,2] => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [6,4,5,3,2,1] => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [6,4,5,3,1,2] => 0
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [6,4,5,2,3,1] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [6,4,5,2,1,3] => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => [6,4,5,1,3,2] => 1
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [6,4,3,5,2,1] => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [6,4,3,5,1,2] => 0
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [6,4,3,2,5,1] => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [6,4,3,2,1,5] => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,2,5] => [6,4,3,1,5,2] => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,2,4,6] => [6,4,2,5,3,1] => 0
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,2,6,4] => [6,4,2,5,1,3] => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,2,4] => [6,4,2,1,5,3] => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,2,4,5] => [6,4,1,5,3,2] => 0
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => [6,3,5,4,2,1] => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => [6,3,5,4,1,2] => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3,6] => [6,3,5,2,4,1] => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => [6,3,5,2,1,4] => 0
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,2,6,3,5] => [6,3,5,1,4,2] => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,2,3,6] => [6,3,2,5,4,1] => 0
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => [6,3,2,5,1,4] => 0
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => [6,3,2,1,5,4] => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => [6,3,1,5,4,2] => 0
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Description
The number of occurrences of the Hertzsprung pattern 132 in a permutation.
A Hertzsprung occurrence of the pattern $\tau=(\tau_1,\dots,\tau_k)$ in a permutation $\pi$ is a factor $\pi_i, \pi_{i+1}, \dots,\pi_{i+k-1}$ of $\pi$ such that $\pi_{i+j-1} - \tau_j$ is constant for $1\leq j\leq k$.
A Hertzsprung occurrence of the pattern $\tau=(\tau_1,\dots,\tau_k)$ in a permutation $\pi$ is a factor $\pi_i, \pi_{i+1}, \dots,\pi_{i+k-1}$ of $\pi$ such that $\pi_{i+j-1} - \tau_j$ is constant for $1\leq j\leq k$.
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
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
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