Identifier
-
Mp00031:
Dyck paths
—to 312-avoiding permutation⟶
Permutations
Mp00309: Permutations —inverse toric promotion⟶ Permutations
St000372: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => 0
[1,0,1,0] => [1,2] => [1,2] => 0
[1,1,0,0] => [2,1] => [2,1] => 0
[1,0,1,0,1,0] => [1,2,3] => [3,2,1] => 0
[1,0,1,1,0,0] => [1,3,2] => [2,3,1] => 0
[1,1,0,0,1,0] => [2,1,3] => [3,1,2] => 0
[1,1,0,1,0,0] => [2,3,1] => [1,3,2] => 0
[1,1,1,0,0,0] => [3,2,1] => [1,2,3] => 1
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [3,1,4,2] => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [3,2,1,4] => 0
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [4,3,2,1] => 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] => [1,4,3,2] => [2,4,3,1] => 0
[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] => [3,2,4,1] => 0
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [3,4,1,2] => 0
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [1,3,4,2] => 1
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [3,1,2,4] => 1
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [4,3,1,2] => 0
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [4,1,3,2] => 0
[1,1,1,0,1,0,0,0] => [3,4,2,1] => [4,1,2,3] => 1
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [2,4,1,3] => 0
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [3,1,4,5,2] => 1
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [3,1,4,2,5] => 2
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [3,1,5,4,2] => 0
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [3,1,5,2,4] => 1
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [3,2,1,5,4] => 0
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [4,3,1,5,2] => 0
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [4,3,2,1,5] => 0
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [4,1,5,3,2] => 0
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [4,1,5,2,3] => 1
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [4,2,1,5,3] => 0
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [5,4,3,2,1] => 0
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [5,4,2,1,3] => 0
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => [5,2,1,4,3] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [2,5,4,3,1] => 0
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [3,4,1,5,2] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [3,4,2,1,5] => 1
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [3,5,4,2,1] => 0
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [3,5,2,1,4] => 0
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [3,2,5,4,1] => 0
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [3,4,5,2,1] => 1
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [3,4,2,5,1] => 1
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [3,4,5,1,2] => 1
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [1,3,4,5,2] => 2
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [3,4,1,2,5] => 2
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [3,5,4,1,2] => 0
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [3,5,1,4,2] => 0
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => [3,5,1,2,4] => 1
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [3,2,5,1,4] => 0
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [4,3,5,2,1] => 0
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [4,3,2,5,1] => 0
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [4,3,5,1,2] => 0
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [4,1,3,5,2] => 1
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [4,3,1,2,5] => 1
[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => [4,5,3,1,2] => 0
[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => [4,5,1,3,2] => 0
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => [4,5,1,2,3] => 1
[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => [4,2,5,1,3] => 0
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [5,4,3,1,2] => 0
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [5,4,1,3,2] => 0
[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => [5,4,1,2,3] => 1
[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => [5,2,4,1,3] => 0
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [2,5,4,1,3] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [3,1,4,5,6,2] => 2
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [3,1,4,5,2,6] => 3
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [3,1,4,6,5,2] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [3,1,4,6,2,5] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [3,1,4,2,6,5] => 2
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [3,1,5,4,6,2] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [3,1,5,4,2,6] => 3
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [3,1,5,6,4,2] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [3,1,5,6,2,4] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => [3,1,5,2,6,4] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => [3,1,6,5,4,2] => 0
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [3,1,6,5,2,4] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => [3,1,6,2,5,4] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [3,2,1,6,5,4] => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [4,3,1,5,6,2] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [4,3,1,5,2,6] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [4,3,1,6,5,2] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [4,3,1,6,2,5] => 1
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [4,3,2,1,6,5] => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [4,1,5,3,6,2] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [4,1,5,3,2,6] => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [4,1,5,6,3,2] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [4,1,5,6,2,3] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => [4,1,5,2,6,3] => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [4,1,6,5,3,2] => 0
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [4,1,6,5,2,3] => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => [4,1,6,2,5,3] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [4,2,1,6,5,3] => 0
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [5,4,3,1,6,2] => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [5,4,3,2,1,6] => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => [5,4,1,6,3,2] => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [5,4,1,6,2,3] => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => [5,4,2,1,6,3] => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => [5,1,6,4,3,2] => 0
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => [5,1,6,4,2,3] => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => [5,1,6,2,4,3] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => [5,2,1,6,4,3] => 0
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Description
The number of mid points of increasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) < \pi(j) < \pi(k)$.
The generating function is given by [1].
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) < \pi(j) < \pi(k)$.
The generating function is given by [1].
Map
inverse toric promotion
Description
Toric promotion of a permutation.
Let $\sigma\in\mathfrak S_n$ be a permutation and let
$ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $
The toric promotion operator is the product $\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$.
This is the special case of toric promotion on graphs for the path graph. Its order is $n-1$.
Let $\sigma\in\mathfrak S_n$ be a permutation and let
$ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $
The toric promotion operator is the product $\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$.
This is the special case of toric promotion on graphs for the path graph. Its order is $n-1$.
Map
to 312-avoiding permutation
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