Identifier
-
Mp00119:
Dyck paths
—to 321-avoiding permutation (Krattenthaler)⟶
Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000663: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => [1] => 0
[1,0,1,0] => [1,2] => [1,2] => [2,1] => 0
[1,1,0,0] => [2,1] => [1,2] => [2,1] => 0
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,0,1,1,0,0] => [1,3,2] => [1,2,3] => [3,2,1] => 0
[1,1,0,0,1,0] => [2,1,3] => [1,2,3] => [3,2,1] => 0
[1,1,0,1,0,0] => [2,3,1] => [1,2,3] => [3,2,1] => 0
[1,1,1,0,0,0] => [3,1,2] => [1,3,2] => [2,3,1] => 0
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,1,1,0,0,0] => [1,4,2,3] => [1,2,4,3] => [3,4,2,1] => 0
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 0
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 0
[1,1,0,1,1,0,0,0] => [2,4,1,3] => [1,2,4,3] => [3,4,2,1] => 0
[1,1,1,0,0,0,1,0] => [3,1,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[1,1,1,0,0,1,0,0] => [3,1,4,2] => [1,3,4,2] => [2,4,3,1] => 1
[1,1,1,0,1,0,0,0] => [3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 0
[1,1,1,1,0,0,0,0] => [4,1,2,3] => [1,4,3,2] => [2,3,4,1] => 0
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [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] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => 0
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => [1,2,3,5,4] => [4,5,3,2,1] => 0
[1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => 0
[1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => 1
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => 0
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => 0
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => [1,2,3,5,4] => [4,5,3,2,1] => 0
[1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => 0
[1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => 1
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => [1,2,4,3,5] => [5,3,4,2,1] => 0
[1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => 0
[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => 0
[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => [1,3,2,4,5] => [5,4,2,3,1] => 0
[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => 1
[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => 1
[1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => [1,3,5,4,2] => [2,4,5,3,1] => 0
[1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => [1,3,2,4,5] => [5,4,2,3,1] => 0
[1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => [1,3,2,4,5] => [5,4,2,3,1] => 0
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => [1,3,5,2,4] => [4,2,5,3,1] => 1
[1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => [1,3,2,5,4] => [4,5,2,3,1] => 0
[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => [1,4,3,2,5] => [5,2,3,4,1] => 0
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => [1,4,5,3,2] => [2,3,5,4,1] => 0
[1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => [1,4,2,3,5] => [5,3,2,4,1] => 1
[1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => [1,4,2,5,3] => [3,5,2,4,1] => 2
[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => [1,2,3,4,6,5] => [5,6,4,3,2,1] => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,3,5] => [1,2,3,4,6,5] => [5,6,4,3,2,1] => 0
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => 0
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => [1,2,3,5,6,4] => [4,6,5,3,2,1] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => 0
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => [1,2,3,6,5,4] => [4,5,6,3,2,1] => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => [1,2,3,4,6,5] => [5,6,4,3,2,1] => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,2,5] => [1,2,3,4,6,5] => [5,6,4,3,2,1] => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,2,4,6] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => 0
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,2,6,4] => [1,2,3,5,6,4] => [4,6,5,3,2,1] => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,2,4] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,2,4,5] => [1,2,3,6,5,4] => [4,5,6,3,2,1] => 0
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3,6] => [1,2,4,5,3,6] => [6,3,5,4,2,1] => 1
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => [1,2,4,5,6,3] => [3,6,5,4,2,1] => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,2,6,3,5] => [1,2,4,6,5,3] => [3,5,6,4,2,1] => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,2,3,6] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => 0
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => 0
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => [1,2,4,6,3,5] => [5,3,6,4,2,1] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => [1,2,4,3,6,5] => [5,6,3,4,2,1] => 0
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Description
The number of right floats of a permutation.
Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, St000441The number of successions of a permutation., and a right float is a large ascent not consecutive to any raft of $\pi$.
See Definition 3.10 and Example 3.11 in [1].
Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, St000441The number of successions of a permutation., and a right float is a large ascent not consecutive to any raft of $\pi$.
See Definition 3.10 and Example 3.11 in [1].
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
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
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
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