Identifier
-
Mp00129:
Dyck paths
—to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶
Permutations
St000124: Permutations ⟶ ℤ
Values
[1,0] => [1] => 1
[1,0,1,0] => [2,1] => 1
[1,1,0,0] => [1,2] => 1
[1,0,1,0,1,0] => [2,3,1] => 1
[1,0,1,1,0,0] => [2,1,3] => 1
[1,1,0,0,1,0] => [1,3,2] => 1
[1,1,0,1,0,0] => [3,1,2] => 2
[1,1,1,0,0,0] => [1,2,3] => 1
[1,0,1,0,1,0,1,0] => [2,3,4,1] => 1
[1,0,1,0,1,1,0,0] => [2,3,1,4] => 1
[1,0,1,1,0,0,1,0] => [2,1,4,3] => 1
[1,0,1,1,0,1,0,0] => [2,4,1,3] => 2
[1,0,1,1,1,0,0,0] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,0] => [1,3,4,2] => 1
[1,1,0,0,1,1,0,0] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0] => [3,1,4,2] => 2
[1,1,0,1,0,1,0,0] => [3,4,1,2] => 2
[1,1,0,1,1,0,0,0] => [3,1,2,4] => 2
[1,1,1,0,0,0,1,0] => [1,2,4,3] => 1
[1,1,1,0,0,1,0,0] => [1,4,2,3] => 2
[1,1,1,0,1,0,0,0] => [4,1,2,3] => 6
[1,1,1,1,0,0,0,0] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => 1
[1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => 1
[1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => 2
[1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => 1
[1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => 1
[1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => 1
[1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => 2
[1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => 2
[1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => 2
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => 1
[1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => 2
[1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => 6
[1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => 1
[1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => 1
[1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => 1
[1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => 2
[1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => 2
[1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => 2
[1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => 2
[1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => 2
[1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => 2
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => 2
[1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => 4
[1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => 6
[1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => 2
[1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => 1
[1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => 1
[1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => 2
[1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => 2
[1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => 2
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => 6
[1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => 6
[1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => 6
[1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => 6
[1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => 1
[1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => 2
[1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => 6
[1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => 24
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,3,4,6,1,5] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,6] => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,3,5,1,6,4] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,3,5,6,1,4] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,3,1,6,4,5] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,3,6,1,4,5] => 6
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6] => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => 1
[1,0,1,1,0,0,1,1,0,1,0,0] => [2,1,4,6,3,5] => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [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] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,4,1,5,3,6] => 2
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,4,5,1,6,3] => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,4,5,6,1,3] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,6,5] => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,4,1,6,3,5] => 4
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,4,6,1,3,5] => 6
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,4,1,3,5,6] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [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
[1,0,1,1,1,0,0,1,0,1,0,0] => [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
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,5,1,3,6,4] => 6
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,5,1,6,3,4] => 6
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,5,6,1,3,4] => 6
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,5,1,3,4,6] => 6
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Description
The cardinality of the preimage of the Simion-Schmidt map.
The Simion-Schmidt bijection transforms a [3,1,2]-avoiding permutation into a [3,2,1]-avoiding permutation. More generally, it can be thought of as a map $S$ that turns any permutation into a [3,2,1]-avoiding permutation. This statistic is the size of $S^{-1}(\pi)$ for each permutation $\pi$.
The map $S$ can also be realized using the quotient of the $0$-Hecke Monoid of the symmetric group by the relation $\pi_i \pi_{i+1} \pi_i = \pi_{i+1} \pi_i$, sending each element of the fiber of the quotient to the unique [3,2,1]-avoiding element in that fiber.
The Simion-Schmidt bijection transforms a [3,1,2]-avoiding permutation into a [3,2,1]-avoiding permutation. More generally, it can be thought of as a map $S$ that turns any permutation into a [3,2,1]-avoiding permutation. This statistic is the size of $S^{-1}(\pi)$ for each permutation $\pi$.
The map $S$ can also be realized using the quotient of the $0$-Hecke Monoid of the symmetric group by the relation $\pi_i \pi_{i+1} \pi_i = \pi_{i+1} \pi_i$, sending each element of the fiber of the quotient to the unique [3,2,1]-avoiding element in that fiber.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
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