Identifier
-
Mp00025:
Dyck paths
—to 132-avoiding permutation⟶
Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000360: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => 0
[1,0,1,0] => [2,1] => [2,1] => 0
[1,1,0,0] => [1,2] => [1,2] => 0
[1,0,1,0,1,0] => [3,2,1] => [3,2,1] => 1
[1,0,1,1,0,0] => [2,3,1] => [3,2,1] => 1
[1,1,0,0,1,0] => [3,1,2] => [3,2,1] => 1
[1,1,0,1,0,0] => [2,1,3] => [2,1,3] => 0
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [4,3,2,1] => [4,3,2,1] => 3
[1,0,1,0,1,1,0,0] => [3,4,2,1] => [4,3,2,1] => 3
[1,0,1,1,0,0,1,0] => [4,2,3,1] => [4,3,2,1] => 3
[1,0,1,1,0,1,0,0] => [3,2,4,1] => [4,2,3,1] => 1
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [4,2,3,1] => 1
[1,1,0,0,1,0,1,0] => [4,3,1,2] => [4,3,2,1] => 3
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [4,3,2,1] => 3
[1,1,0,1,0,0,1,0] => [4,2,1,3] => [4,3,2,1] => 3
[1,1,0,1,0,1,0,0] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,0,1,1,0,0,0] => [2,3,1,4] => [3,2,1,4] => 1
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [4,2,3,1] => 1
[1,1,1,0,0,1,0,0] => [3,1,2,4] => [3,2,1,4] => 1
[1,1,1,0,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => 0
[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] => [5,4,3,2,1] => [5,4,3,2,1] => 6
[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [5,4,3,2,1] => 6
[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [5,4,3,2,1] => 6
[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => [5,4,3,2,1] => 6
[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [5,4,3,2,1] => 6
[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [5,4,3,2,1] => 6
[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [5,4,3,2,1] => 6
[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => [5,4,3,2,1] => 6
[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => [5,3,2,4,1] => 3
[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => [5,3,2,4,1] => 3
[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [5,4,3,2,1] => 6
[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => [5,3,2,4,1] => 3
[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => [5,2,3,4,1] => 1
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [5,4,3,2,1] => 6
[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [5,4,3,2,1] => 6
[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [5,4,3,2,1] => 6
[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => [5,4,3,2,1] => 6
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [5,4,3,2,1] => 6
[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => [5,4,3,2,1] => 6
[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => [5,4,3,2,1] => 6
[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => [5,4,3,2,1] => 6
[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => [4,3,2,1,5] => 3
[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => [4,3,2,1,5] => 3
[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => [5,4,3,2,1] => 6
[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => [4,3,2,1,5] => 3
[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => [4,2,3,1,5] => 1
[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => [4,2,3,1,5] => 1
[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => [5,4,3,2,1] => 6
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [5,4,3,2,1] => 6
[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => [5,4,3,2,1] => 6
[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => [4,3,2,1,5] => 3
[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => [4,3,2,1,5] => 3
[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => [5,3,2,4,1] => 3
[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => [4,3,2,1,5] => 3
[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => [3,2,1,4,5] => 1
[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [5,2,3,4,1] => 1
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => [4,2,3,1,5] => 1
[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => [3,2,1,4,5] => 1
[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[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] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 10
[1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => [6,5,4,3,2,1] => 10
[1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => [6,5,4,3,2,1] => 10
[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => [6,5,4,3,2,1] => 10
[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [6,5,4,3,2,1] => 10
[1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => 10
[1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [6,5,4,3,2,1] => 10
[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => [6,5,4,3,2,1] => 10
[1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => [6,5,3,4,2,1] => 7
[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => [6,5,3,4,2,1] => 7
[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [6,5,4,3,2,1] => 10
[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => [6,5,3,4,2,1] => 7
[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => [6,5,3,4,2,1] => 7
[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [6,5,3,4,2,1] => 7
[1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => [6,5,4,3,2,1] => 10
[1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [6,5,4,3,2,1] => 10
[1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [6,5,4,3,2,1] => 10
[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => [6,5,4,3,2,1] => 10
[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [6,5,4,3,2,1] => 10
[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => [6,5,4,3,2,1] => 10
[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => [6,5,4,3,2,1] => 10
[1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => [6,5,4,3,2,1] => 10
[1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => [6,4,3,2,5,1] => 6
[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => [6,4,3,2,5,1] => 6
[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => [6,5,4,3,2,1] => 10
[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => [6,4,3,2,5,1] => 6
[1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => [6,4,3,2,5,1] => 6
[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => [6,4,3,2,5,1] => 6
[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [6,5,4,3,2,1] => 10
[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [6,5,4,3,2,1] => 10
[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => [6,5,4,3,2,1] => 10
[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => [6,4,3,2,5,1] => 6
[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => [6,4,3,2,5,1] => 6
[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => [6,5,3,4,2,1] => 7
[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [6,4,3,2,5,1] => 6
[1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => [6,3,2,4,5,1] => 3
[1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => [6,3,2,4,5,1] => 3
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Description
The number of occurrences of the pattern 32-1.
See Permutations/#Pattern-avoiding_permutations for the definition of the pattern $32\!\!-\!\!1$.
See Permutations/#Pattern-avoiding_permutations for the definition of the pattern $32\!\!-\!\!1$.
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 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
This bijection is defined in [1, Section 2].
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