Identifier
-
Mp00129:
Dyck paths
—to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶
Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001737: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => 0
[1,0,1,0] => [2,1] => [2,1] => 1
[1,1,0,0] => [1,2] => [1,2] => 0
[1,0,1,0,1,0] => [2,3,1] => [3,1,2] => 1
[1,0,1,1,0,0] => [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,0] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0] => [3,1,2] => [2,3,1] => 1
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [2,3,4,1] => [4,1,2,3] => 1
[1,0,1,0,1,1,0,0] => [2,3,1,4] => [3,1,2,4] => 1
[1,0,1,1,0,0,1,0] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0] => [2,4,1,3] => [3,4,1,2] => 1
[1,0,1,1,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,0] => [1,3,4,2] => [1,4,2,3] => 1
[1,1,0,0,1,1,0,0] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0] => [3,1,4,2] => [4,2,3,1] => 1
[1,1,0,1,0,1,0,0] => [3,4,1,2] => [2,4,1,3] => 1
[1,1,0,1,1,0,0,0] => [3,1,2,4] => [2,3,1,4] => 1
[1,1,1,0,0,0,1,0] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,0,0,1,0,0] => [1,4,2,3] => [1,3,4,2] => 1
[1,1,1,0,1,0,0,0] => [4,1,2,3] => [2,3,4,1] => 1
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => [4,1,2,3,5] => 1
[1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => [3,1,2,5,4] => 2
[1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => [4,5,1,2,3] => 1
[1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => [3,1,2,4,5] => 1
[1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => [2,1,5,3,4] => 2
[1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => [5,3,4,1,2] => 1
[1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => [3,5,1,2,4] => 1
[1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => [3,4,1,2,5] => 1
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => [2,1,4,5,3] => 2
[1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => [3,4,5,1,2] => 1
[1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => [1,4,5,2,3] => 1
[1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => [5,2,3,1,4] => 1
[1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => [4,2,3,1,5] => 1
[1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => [5,2,4,1,3] => 1
[1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => [2,5,1,3,4] => 1
[1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => [2,4,1,3,5] => 1
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => [2,3,1,5,4] => 2
[1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => [4,5,2,3,1] => 1
[1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => [2,4,5,1,3] => 1
[1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => [2,3,1,4,5] => 1
[1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => [1,5,3,4,2] => 1
[1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => [1,3,5,2,4] => 1
[1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => [2,5,3,4,1] => 1
[1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => [3,5,2,4,1] => 1
[1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => [2,3,5,1,4] => 1
[1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => [2,3,4,1,5] => 1
[1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => [1,3,4,5,2] => 1
[1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => [2,3,4,5,1] => 1
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => [5,1,2,3,4,6] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => [4,1,2,3,6,5] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,3,4,6,1,5] => [5,6,1,2,3,4] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => [4,1,2,3,5,6] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => [3,1,2,6,4,5] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,6] => [3,1,2,5,4,6] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,3,5,1,6,4] => [6,4,5,1,2,3] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,3,5,6,1,4] => [4,6,1,2,3,5] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => [4,5,1,2,3,6] => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => [3,1,2,4,6,5] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,3,1,6,4,5] => [3,1,2,5,6,4] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,3,6,1,4,5] => [4,5,6,1,2,3] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => [3,1,2,4,5,6] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => [2,1,6,3,4,5] => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6] => [2,1,5,3,4,6] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3
[1,0,1,1,0,0,1,1,0,1,0,0] => [2,1,4,6,3,5] => [2,1,5,6,3,4] => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,4,1,5,6,3] => [6,3,4,1,2,5] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,4,1,5,3,6] => [5,3,4,1,2,6] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,4,5,1,6,3] => [6,3,5,1,2,4] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,4,5,6,1,3] => [3,6,1,2,4,5] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => [3,5,1,2,4,6] => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,6,5] => [3,4,1,2,6,5] => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,4,1,6,3,5] => [5,6,3,4,1,2] => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,4,6,1,3,5] => [3,5,6,1,2,4] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,4,1,3,5,6] => [3,4,1,2,5,6] => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => [2,1,3,6,4,5] => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,3,6,4] => [2,1,6,4,5,3] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [2,1,5,6,3,4] => [2,1,4,6,3,5] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [2,1,5,3,4,6] => [2,1,4,5,3,6] => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,5,1,3,6,4] => [3,6,4,5,1,2] => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,5,1,6,3,4] => [4,6,3,5,1,2] => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,5,6,1,3,4] => [3,4,6,1,2,5] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,5,1,3,4,6] => [3,4,5,1,2,6] => 1
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Description
The number of descents of type 2 in a permutation.
A position $i\in[1,n-1]$ is a descent of type 2 of a permutation $\pi$ of $n$ letters, if it is a descent and if $\pi(j) < \pi(i)$ for all $j < i$.
A position $i\in[1,n-1]$ is a descent of type 2 of a permutation $\pi$ of $n$ letters, if it is a descent and if $\pi(j) < \pi(i)$ for all $j < i$.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
Map
invert Laguerre heap
Description
The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.
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