Identifier
-
Mp00230:
Integer partitions
—parallelogram polyomino⟶
Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000124: Permutations ⟶ ℤ
Values
[1] => [1,0] => [1] => 1
[2] => [1,0,1,0] => [2,1] => 1
[1,1] => [1,1,0,0] => [1,2] => 1
[3] => [1,0,1,0,1,0] => [2,1,3] => 1
[2,1] => [1,0,1,1,0,0] => [2,3,1] => 1
[1,1,1] => [1,1,0,1,0,0] => [1,3,2] => 1
[4] => [1,0,1,0,1,0,1,0] => [2,1,4,3] => 1
[3,1] => [1,0,1,0,1,1,0,0] => [2,4,1,3] => 2
[2,2] => [1,1,1,0,0,0] => [1,2,3] => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => [2,3,1,4] => 1
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,3,2,4] => 1
[5] => [1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => 1
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => 2
[3,2] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [2,4,1,5,3] => 2
[2,2,1] => [1,1,1,0,0,1,0,0] => [1,4,2,3] => 2
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [2,3,1,5,4] => 1
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4] => 1
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5] => 1
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,6,5] => 2
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => 2
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [2,4,1,6,3,5] => 4
[3,3] => [1,1,1,0,1,0,0,0] => [1,2,4,3] => 1
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [2,3,1,4,5] => 1
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [2,4,1,5,3,6] => 2
[2,2,2] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 1
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,3] => 2
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [2,3,1,5,4,6] => 1
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4,6] => 1
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => 6
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [2,3,4,1,5] => 1
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [2,4,1,5,6,3] => 2
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,4,2,3,5] => 2
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [2,3,1,6,4,5] => 2
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => 6
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,2,5,3,6] => 2
[8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5,8,7] => 1
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [2,4,5,1,6,3] => 2
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,2,4,3,5] => 1
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [2,3,1,4,6,5] => 1
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => 2
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => 1
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [1,4,2,6,3,5] => 4
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [2,3,1,4,5,6] => 1
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => 1
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,5,2,6,3,4] => 6
[1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4,7,6,8] => 1
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [2,3,4,1,6,5] => 1
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [1,4,2,3,6,5] => 2
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [2,3,4,6,1,5] => 2
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 1
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [1,4,2,3,5,6] => 2
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [2,3,4,5,1,6] => 1
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,5,2,3,4,6] => 6
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,2,4,3,6,5] => 1
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [1,2,4,6,3,5] => 2
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 1
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,6,2,3,4,5] => 24
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,2,4,5,3,6] => 1
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,2,3,5,4,6] => 1
[2,2,2,2,1,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0] => [1,5,2,7,3,4,6] => 18
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,2,4,5,6,3] => 1
[4,3,3,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [2,3,1,4,5,6,7] => 1
[4,3,2,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [2,3,4,6,1,7,5] => 2
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,2,3,6,4,5] => 2
[3,3,3,1,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [1,6,2,7,3,4,5] => 24
[3,2,2,2,2] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [2,3,4,5,1,7,6] => 1
[2,2,2,2,2,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0] => [1,5,2,3,4,7,6] => 6
[5,4,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [2,3,4,6,7,1,5] => 2
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,2,3,4,6,5] => 1
[4,3,3,2] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [2,3,4,5,1,6,7] => 1
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 1
[3,3,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0] => [1,6,2,3,4,7,5] => 24
[5,4,4] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [2,3,4,5,6,1,7] => 1
[4,3,3,3] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => 1
[3,3,3,3,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [1,7,2,3,4,5,6] => 120
[3,3,3,2,2] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [1,2,3,6,4,7,5] => 2
[2,2,2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0] => [1,2,3,5,4,7,6,8] => 1
[5,5,5] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [1,2,3,4,6,5,7] => 1
[4,3,3,3,2] => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [2,3,4,5,1,6,7,8] => 1
[3,3,3,3,3] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1,2,3,4,5,7,6] => 1
[6,5,5] => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,3,4,5,6,1,8,7] => 1
[4,4,4,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => 1
[4,3,3,3,3] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1,8] => 1
[5,4,4,4] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => 1
[3,3,3,3,3,3] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [1,2,3,4,5,7,6,8] => 1
[4,4,4,4,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7,8] => 1
[5,5,5,5] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,8,7] => 1
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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
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
to 321-avoiding permutation
Description
Sends a Dyck path to a 321-avoiding permutation.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.
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