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Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000124
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Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000124: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000124: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,2] => 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 6
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 6
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 6
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 6
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => 6
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 6
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 24
[1,1,1,1,0,0,0,0]
=> [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]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [3,4,1,6,2,5] => 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,2,5] => 6
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,1,2,5,6] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [3,1,5,6,2,4] => 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4,6] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [3,5,1,6,2,4] => 6
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,5,6,1,2,4] => 6
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4,6] => 6
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [3,1,2,6,4,5] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,6,2,4,5] => 12
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,6,1,2,4,5] => 24
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,6,2,3] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,3,6] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,6,3,5] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,6,2,3,5] => 6
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,3] => 6
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,1,5,2,3,6] => 6
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,5,1,6,2,3] => 6
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,1,2,3] => 6
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6] => 6
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,2,6,3,5] => 12
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,1,6,2,3,5] => 18
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,6,1,2,3,5] => 24
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,1,2,3,5,6] => 6
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.
Matching statistic: St001232
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 18%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 18%
Values
[1,0]
=> [1] => [1]
=> [1,0,1,0]
=> 1
[1,0,1,0]
=> [2,1] => [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0]
=> [1,2] => [2]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> ? = 2
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> ? = 2
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> [1,0,1,0,1,0]
=> ? = 2
[1,1,0,1,0,0]
=> [1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> ? = 6
[1,1,1,0,0,0]
=> [1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 4
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 6
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 2
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 6
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 6
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 6
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 6
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 24
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 4
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 6
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 4
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 4
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 6
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 6
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 6
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 4
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 12
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 24
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 2
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 4
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 6
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 6
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 6
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 6
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 6
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 6
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 12
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 18
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 24
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 6
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 6
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 6
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 6
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 24
[1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 24
[1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 24
[1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 24
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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