Identifier
-
Mp00129:
Dyck paths
—to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶
Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000516: Permutations ⟶ ℤ
Values
[1,0,1,0] => [2,1] => [2,1] => [2,1] => 0
[1,1,0,0] => [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0] => [2,3,1] => [3,1,2] => [3,2,1] => 0
[1,0,1,1,0,0] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[1,1,0,0,1,0] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[1,1,0,1,0,0] => [3,1,2] => [2,3,1] => [3,2,1] => 0
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [2,3,4,1] => [4,1,2,3] => [4,2,3,1] => 0
[1,0,1,0,1,1,0,0] => [2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 0
[1,0,1,1,0,0,1,0] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
[1,0,1,1,0,1,0,0] => [2,4,1,3] => [3,4,1,2] => [4,3,2,1] => 0
[1,0,1,1,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[1,1,0,0,1,0,1,0] => [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 0
[1,1,0,0,1,1,0,0] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,1,0,1,0,0,1,0] => [3,1,4,2] => [4,2,3,1] => [4,3,2,1] => 0
[1,1,0,1,0,1,0,0] => [3,4,1,2] => [2,4,1,3] => [3,4,1,2] => 0
[1,1,0,1,1,0,0,0] => [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 0
[1,1,1,0,0,0,1,0] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,1,1,0,0,1,0,0] => [1,4,2,3] => [1,3,4,2] => [1,4,3,2] => 0
[1,1,1,0,1,0,0,0] => [4,1,2,3] => [2,3,4,1] => [4,2,3,1] => 0
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [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] => [5,2,3,4,1] => 0
[1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => [4,1,2,3,5] => [4,2,3,1,5] => 0
[1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => [3,1,2,5,4] => [3,2,1,5,4] => 1
[1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => [4,5,1,2,3] => [5,4,3,2,1] => 0
[1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 0
[1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,4,3] => 1
[1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 1
[1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => [5,3,4,1,2] => [5,4,3,2,1] => 0
[1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => [3,5,1,2,4] => [4,5,3,1,2] => 0
[1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => [3,4,1,2,5] => [4,3,2,1,5] => 0
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 1
[1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 1
[1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => [3,4,5,1,2] => [5,4,3,2,1] => 0
[1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => [1,5,2,3,4] => [1,5,3,4,2] => 0
[1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => [1,4,2,3,5] => [1,4,3,2,5] => 0
[1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 1
[1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => [1,4,5,2,3] => [1,5,4,3,2] => 0
[1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => [5,2,3,1,4] => [5,4,3,2,1] => 0
[1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => [4,2,3,1,5] => [4,3,2,1,5] => 0
[1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => [5,2,4,1,3] => [5,4,3,2,1] => 0
[1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => [2,5,1,3,4] => [3,5,1,4,2] => 0
[1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => [2,4,1,3,5] => [3,4,1,2,5] => 0
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => [2,3,1,5,4] => [3,2,1,5,4] => 1
[1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => [4,5,2,3,1] => [5,4,3,2,1] => 0
[1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => [2,4,5,1,3] => [4,5,3,1,2] => 0
[1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => [2,3,1,4,5] => [3,2,1,4,5] => 0
[1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,4,3] => 0
[1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => [1,5,3,4,2] => [1,5,4,3,2] => 0
[1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => [1,3,5,2,4] => [1,4,5,2,3] => 0
[1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => 0
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => [2,5,3,4,1] => [5,4,3,2,1] => 0
[1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => [3,5,2,4,1] => [5,4,3,2,1] => 0
[1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => [2,3,5,1,4] => [4,2,5,1,3] => 0
[1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => 0
[1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,4,3] => 0
[1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => [1,3,4,5,2] => [1,5,3,4,2] => 0
[1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => [2,3,4,5,1] => [5,2,3,4,1] => 0
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [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] => [6,2,3,4,5,1] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => [5,1,2,3,4,6] => [5,2,3,4,1,6] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => [4,1,2,3,6,5] => [4,2,3,1,6,5] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,3,4,6,1,5] => [5,6,1,2,3,4] => [6,5,3,4,2,1] => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => [4,1,2,3,5,6] => [4,2,3,1,5,6] => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => [3,1,2,6,4,5] => [3,2,1,6,5,4] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,6] => [3,1,2,5,4,6] => [3,2,1,5,4,6] => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,3,5,1,6,4] => [6,4,5,1,2,3] => [6,5,4,3,2,1] => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,3,5,6,1,4] => [4,6,1,2,3,5] => [5,6,3,4,1,2] => 0
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => [4,5,1,2,3,6] => [5,4,3,2,1,6] => 0
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => [3,1,2,4,6,5] => [3,2,1,4,6,5] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,3,1,6,4,5] => [3,1,2,5,6,4] => [3,2,1,6,5,4] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,3,6,1,4,5] => [4,5,6,1,2,3] => [6,5,4,3,2,1] => 0
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => [3,1,2,4,5,6] => [3,2,1,4,5,6] => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => [2,1,6,3,4,5] => [2,1,6,4,5,3] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6] => [2,1,5,3,4,6] => [2,1,5,4,3,6] => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => [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] => [2,1,6,5,4,3] => 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] => [2,1,4,3,5,6] => 1
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,4,1,5,6,3] => [6,3,4,1,2,5] => [6,5,4,3,2,1] => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,4,1,5,3,6] => [5,3,4,1,2,6] => [5,4,3,2,1,6] => 0
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,4,5,1,6,3] => [6,3,5,1,2,4] => [6,5,4,3,2,1] => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,4,5,6,1,3] => [3,6,1,2,4,5] => [4,6,3,1,5,2] => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => [3,5,1,2,4,6] => [4,5,3,1,2,6] => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,6,5] => [3,4,1,2,6,5] => [4,3,2,1,6,5] => 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] => [6,5,4,3,2,1] => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,4,6,1,3,5] => [3,5,6,1,2,4] => [5,6,4,3,1,2] => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,4,1,3,5,6] => [3,4,1,2,5,6] => [4,3,2,1,5,6] => 0
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => [2,1,3,6,4,5] => [2,1,3,6,5,4] => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,3,6,4] => [2,1,6,4,5,3] => [2,1,6,5,4,3] => 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] => [2,1,5,6,3,4] => 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] => [2,1,5,4,3,6] => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,5,1,3,6,4] => [3,6,4,5,1,2] => [6,5,4,3,2,1] => 0
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,5,1,6,3,4] => [4,6,3,5,1,2] => [6,5,4,3,2,1] => 0
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,5,6,1,3,4] => [3,4,6,1,2,5] => [5,4,6,2,1,3] => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,5,1,3,4,6] => [3,4,5,1,2,6] => [5,4,3,2,1,6] => 0
[1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => 1
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Description
The number of stretching pairs of a permutation.
This is the number of pairs $(i,j)$ with $\pi(i) < i < j < \pi(j)$.
This is the number of pairs $(i,j)$ with $\pi(i) < i < j < \pi(j)$.
Map
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
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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