searching the database
Your data matches 28 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000214
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => 0
[1,2] => [2] => [1,1,0,0]
=> [1,2] => 0
[2,1] => [1,1] => [1,0,1,0]
=> [2,1] => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 0
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
Description
The number of adjacencies of a permutation.
An adjacency of a permutation π is an index i such that π(i)−1=π(i+1). Adjacencies are also known as ''small descents''.
This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Matching statistic: St001465
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => 0
[1,2] => [2] => [1,1,0,0]
=> [1,2] => 0
[2,1] => [1,1] => [1,0,1,0]
=> [2,1] => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 0
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
Description
The number of adjacent transpositions in the cycle decomposition of a permutation.
Matching statistic: St000683
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000683: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 90%●distinct values known / distinct values provided: 71%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000683: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 90%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1,0]
=> [1,0]
=> ? = 0
[1,2] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,7,6,4,5,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 0
[8,7,5,4,6,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 0
[8,6,5,4,7,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 0
[7,6,5,4,8,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 0
[8,7,6,5,4,2,3,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1
[8,7,6,5,3,2,4,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1
[8,7,6,4,3,2,5,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1
[8,7,5,4,3,2,6,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1
[8,6,5,4,3,2,7,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1
[7,6,5,4,3,2,8,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1
[8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 0
[8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 0
[8,7,6,5,4,1,2,3] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,7,6,5,3,2,1,4] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 0
[8,7,6,5,3,1,2,4] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,7,6,5,2,1,3,4] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,7,6,5,1,2,3,4] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,6,4,3,2,1,5] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 0
[8,7,6,4,3,1,2,5] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,7,6,4,2,1,3,5] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,7,6,4,1,2,3,5] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,6,3,2,1,4,5] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,7,6,3,1,2,4,5] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,6,2,1,3,4,5] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,5,4,3,2,1,6] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 0
[8,7,5,4,3,1,2,6] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,7,5,4,2,1,3,6] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,7,5,4,1,2,3,6] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,5,3,2,1,4,6] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,7,5,3,1,2,4,6] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,5,2,1,3,4,6] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,4,3,2,1,5,6] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,7,4,3,1,2,5,6] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,4,2,1,3,5,6] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,3,2,1,4,5,6] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,6,5,4,3,2,1,7] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 0
[8,6,5,4,3,1,2,7] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,6,5,4,2,1,3,7] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,6,5,4,1,2,3,7] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,6,5,3,2,1,4,7] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,6,5,3,1,2,4,7] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,6,5,2,1,3,4,7] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,6,4,3,2,1,5,7] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,6,4,3,1,2,5,7] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,6,4,2,1,3,5,7] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,6,3,2,1,4,5,7] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,5,4,3,2,1,6,7] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,5,4,3,1,2,6,7] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
Description
The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps.
Matching statistic: St001139
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001139: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 90%●distinct values known / distinct values provided: 71%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001139: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 90%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1,0]
=> [1,0]
=> ? = 0
[1,2] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,7,6,4,5,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 0
[8,7,5,4,6,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 0
[8,6,5,4,7,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 0
[7,6,5,4,8,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 0
[8,7,6,5,4,2,3,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1
[8,7,6,5,3,2,4,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1
[8,7,6,4,3,2,5,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1
[8,7,5,4,3,2,6,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1
[8,6,5,4,3,2,7,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1
[7,6,5,4,3,2,8,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1
[8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 0
[8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 0
[8,7,6,5,4,1,2,3] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,7,6,5,3,2,1,4] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 0
[8,7,6,5,3,1,2,4] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,7,6,5,2,1,3,4] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,7,6,5,1,2,3,4] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,6,4,3,2,1,5] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 0
[8,7,6,4,3,1,2,5] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,7,6,4,2,1,3,5] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,7,6,4,1,2,3,5] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,6,3,2,1,4,5] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,7,6,3,1,2,4,5] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,6,2,1,3,4,5] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,5,4,3,2,1,6] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 0
[8,7,5,4,3,1,2,6] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,7,5,4,2,1,3,6] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,7,5,4,1,2,3,6] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,5,3,2,1,4,6] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,7,5,3,1,2,4,6] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,5,2,1,3,4,6] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,4,3,2,1,5,6] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,7,4,3,1,2,5,6] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,4,2,1,3,5,6] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,3,2,1,4,5,6] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,6,5,4,3,2,1,7] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 0
[8,6,5,4,3,1,2,7] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,6,5,4,2,1,3,7] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,6,5,4,1,2,3,7] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,6,5,3,2,1,4,7] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,6,5,3,1,2,4,7] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,6,5,2,1,3,4,7] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,6,4,3,2,1,5,7] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,6,4,3,1,2,5,7] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,6,4,2,1,3,5,7] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,6,3,2,1,4,5,7] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,5,4,3,2,1,6,7] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[8,5,4,3,1,2,6,7] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
Description
The number of occurrences of hills of size 2 in a Dyck path.
A hill of size two is a subpath beginning at height zero, consisting of two up steps followed by two down steps.
Matching statistic: St001657
Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 0
[1,2] => [2,1] => [1,1] => [1,1]
=> 0
[2,1] => [1,2] => [2] => [2]
=> 1
[1,2,3] => [3,2,1] => [1,1,1] => [1,1,1]
=> 0
[1,3,2] => [2,3,1] => [2,1] => [2,1]
=> 1
[2,1,3] => [3,1,2] => [1,2] => [2,1]
=> 1
[2,3,1] => [1,3,2] => [2,1] => [2,1]
=> 1
[3,1,2] => [2,1,3] => [1,2] => [2,1]
=> 1
[3,2,1] => [1,2,3] => [3] => [3]
=> 0
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => [1,1,1,1]
=> 0
[1,2,4,3] => [3,4,2,1] => [2,1,1] => [2,1,1]
=> 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => [2,1,1]
=> 1
[1,3,4,2] => [2,4,3,1] => [2,1,1] => [2,1,1]
=> 1
[1,4,2,3] => [3,2,4,1] => [1,2,1] => [2,1,1]
=> 1
[1,4,3,2] => [2,3,4,1] => [3,1] => [3,1]
=> 0
[2,1,3,4] => [4,3,1,2] => [1,1,2] => [2,1,1]
=> 1
[2,1,4,3] => [3,4,1,2] => [2,2] => [2,2]
=> 2
[2,3,1,4] => [4,1,3,2] => [1,2,1] => [2,1,1]
=> 1
[2,3,4,1] => [1,4,3,2] => [2,1,1] => [2,1,1]
=> 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [2,1,1]
=> 1
[2,4,3,1] => [1,3,4,2] => [3,1] => [3,1]
=> 0
[3,1,2,4] => [4,2,1,3] => [1,1,2] => [2,1,1]
=> 1
[3,1,4,2] => [2,4,1,3] => [2,2] => [2,2]
=> 2
[3,2,1,4] => [4,1,2,3] => [1,3] => [3,1]
=> 0
[3,2,4,1] => [1,4,2,3] => [2,2] => [2,2]
=> 2
[3,4,1,2] => [2,1,4,3] => [1,2,1] => [2,1,1]
=> 1
[3,4,2,1] => [1,2,4,3] => [3,1] => [3,1]
=> 0
[4,1,2,3] => [3,2,1,4] => [1,1,2] => [2,1,1]
=> 1
[4,1,3,2] => [2,3,1,4] => [2,2] => [2,2]
=> 2
[4,2,1,3] => [3,1,2,4] => [1,3] => [3,1]
=> 0
[4,2,3,1] => [1,3,2,4] => [2,2] => [2,2]
=> 2
[4,3,1,2] => [2,1,3,4] => [1,3] => [3,1]
=> 0
[4,3,2,1] => [1,2,3,4] => [4] => [4]
=> 0
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => [1,1,1,1,1]
=> 0
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => [2,1,1,1]
=> 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => [2,1,1,1]
=> 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => [2,1,1,1]
=> 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => [2,1,1,1]
=> 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => [3,1,1]
=> 0
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => [2,1,1,1]
=> 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => [2,2,1]
=> 2
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => [2,1,1,1]
=> 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => [2,1,1,1]
=> 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => [2,1,1,1]
=> 1
[1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => [3,1,1]
=> 0
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => [2,1,1,1]
=> 1
[1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => [2,2,1]
=> 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => [3,1,1]
=> 0
[1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => [2,2,1]
=> 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => [2,1,1,1]
=> 1
[7,8,4,3,5,2,1,6] => [6,1,2,5,3,4,8,7] => ? => ?
=> ? = 0
[8,7,4,2,1,3,5,6] => [6,5,3,1,2,4,7,8] => ? => ?
=> ? = 0
[8,6,5,3,1,2,4,7] => [7,4,2,1,3,5,6,8] => ? => ?
=> ? = 0
[8,5,4,3,1,2,6,7] => [7,6,2,1,3,4,5,8] => ? => ?
=> ? = 0
[6,5,4,2,1,3,7,8] => [8,7,3,1,2,4,5,6] => ? => ?
=> ? = 0
[2,3,7,8,6,5,4,1] => [1,4,5,6,8,7,3,2] => ? => ?
=> ? = 0
[2,7,3,5,4,6,1,8] => ? => ? => ?
=> ? = 3
[6,2,4,3,5,1,8,7] => [7,8,1,5,3,4,2,6] => ? => ?
=> ? = 4
[5,3,2,4,6,8,7,1] => ? => ? => ?
=> ? = 0
[8,3,4,2,6,5,7,1] => [1,7,5,6,2,4,3,8] => ? => ?
=> ? = 4
[2,4,5,8,1,3,6,7] => [7,6,3,1,8,5,4,2] => ? => ?
=> ? = 1
[2,6,1,3,7,8,4,5] => [5,4,8,7,3,1,6,2] => ? => ?
=> ? = 2
[2,3,7,1,8,4,5,6] => [6,5,4,8,1,7,3,2] => ? => ?
=> ? = 2
[3,4,5,1,8,2,6,7] => [7,6,2,8,1,5,4,3] => ? => ?
=> ? = 2
[1,3,2,4,6,8,5,7] => [7,5,8,6,4,2,3,1] => ? => ?
=> ? = 2
[1,4,5,7,2,8,3,6] => [6,3,8,2,7,5,4,1] => ? => ?
=> ? = 2
[5,6,1,2,3,7,4,8] => [8,4,7,3,2,1,6,5] => ? => ?
=> ? = 2
[1,5,6,2,7,3,4,8] => [8,4,3,7,2,6,5,1] => ? => ?
=> ? = 2
[1,5,6,8,2,3,4,7] => [7,4,3,2,8,6,5,1] => ? => ?
=> ? = 1
[1,5,2,3,6,7,4,8] => [8,4,7,6,3,2,5,1] => ? => ?
=> ? = 2
[1,2,6,7,3,8,4,5] => [5,4,8,3,7,6,2,1] => ? => ?
=> ? = 2
[4,7,2,6,1,3,5,8] => [8,5,3,1,6,2,7,4] => ? => ?
=> ? = 2
[4,8,2,5,1,3,6,7] => [7,6,3,1,5,2,8,4] => ? => ?
=> ? = 2
[4,8,3,5,1,2,6,7] => [7,6,2,1,5,3,8,4] => ? => ?
=> ? = 2
[4,2,7,1,6,3,8,5] => ? => ? => ?
=> ? = 4
[3,6,5,8,1,2,4,7] => [7,4,2,1,8,5,6,3] => ? => ?
=> ? = 2
[3,7,8,1,6,2,4,5] => [5,4,2,6,1,8,7,3] => ? => ?
=> ? = 2
[3,7,1,8,4,6,2,5] => [5,2,6,4,8,1,7,3] => ? => ?
=> ? = 3
[2,6,1,8,5,7,3,4] => [4,3,7,5,8,1,6,2] => ? => ?
=> ? = 3
[5,4,8,3,6,1,7,2] => [2,7,1,6,3,8,4,5] => ? => ?
=> ? = 4
[4,5,1,7,3,6,2,8] => ? => ? => ?
=> ? = 3
[4,7,2,5,1,8,3,6] => ? => ? => ?
=> ? = 3
[4,2,6,3,8,1,7,5] => [5,7,1,8,3,6,2,4] => ? => ?
=> ? = 4
[5,6,8,1,3,2,4,7] => [7,4,2,3,1,8,6,5] => ? => ?
=> ? = 2
[3,1,7,5,6,4,8,2] => [2,8,4,6,5,7,1,3] => ? => ?
=> ? = 4
[2,7,1,4,5,8,3,6] => [6,3,8,5,4,1,7,2] => ? => ?
=> ? = 2
[1,6,4,8,5,7,2,3] => [3,2,7,5,8,4,6,1] => ? => ?
=> ? = 3
[6,7,4,5,1,8,2,3] => ? => ? => ?
=> ? = 3
[2,8,1,4,3,7,5,6] => [6,5,7,3,4,1,8,2] => ? => ?
=> ? = 3
[1,4,3,2,8,6,5,7] => [7,5,6,8,2,3,4,1] => ? => ?
=> ? = 0
[1,3,2,7,5,8,4,6] => [6,4,8,5,7,2,3,1] => ? => ?
=> ? = 3
[1,6,7,8,2,5,3,4] => [4,3,5,2,8,7,6,1] => ? => ?
=> ? = 2
[1,4,8,6,7,2,3,5] => [5,3,2,7,6,8,4,1] => ? => ?
=> ? = 2
[1,2,8,5,7,3,4,6] => [6,4,3,7,5,8,2,1] => ? => ?
=> ? = 2
[10,7,9,5,8,3,6,1,4,2] => ? => ? => ?
=> ? = 5
[10,8,9,5,7,3,6,1,4,2] => ? => ? => ?
=> ? = 5
[10,8,9,6,7,3,5,1,4,2] => ? => ? => ?
=> ? = 5
[5,6,1,7,3,4,2,8] => ? => ? => ?
=> ? = 3
[4,7,2,1,8,5,3,6] => [6,3,5,8,1,2,7,4] => ? => ?
=> ? = 0
[2,6,4,1,7,5,3,8] => [8,3,5,7,1,4,6,2] => ? => ?
=> ? = 0
Description
The number of twos in an integer partition.
The total number of twos in all partitions of n is equal to the total number of singletons [[St001484]] in all partitions of n−1, see [1].
Matching statistic: St000658
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 83%●distinct values known / distinct values provided: 71%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 83%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1] => [1,0]
=> ? = 0
[1,2] => [2,1] => [1,1] => [1,0,1,0]
=> 0
[2,1] => [1,2] => [2] => [1,1,0,0]
=> 1
[1,2,3] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1,3] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,1,2] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,4,2,3] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[2,1,3,4] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,1,4,3] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,3,4,1] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,4,3,1] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[3,1,2,4] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[3,2,4,1] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,4,2,1] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[4,1,2,3] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[4,1,3,2] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,2,1,3] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[4,2,3,1] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,3,1,2] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[4,3,2,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,5,3,2] => [2,3,5,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[5,6,7,8,4,3,2,1] => [1,2,3,4,8,7,6,5] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,6,4,5,3,2,1] => [1,2,3,5,4,6,7,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[8,7,5,4,6,3,2,1] => [1,2,3,6,4,5,7,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[8,6,5,4,7,3,2,1] => [1,2,3,7,4,5,6,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[7,6,5,4,8,3,2,1] => [1,2,3,8,4,5,6,7] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[8,7,3,4,5,6,2,1] => [1,2,6,5,4,3,7,8] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
[8,6,3,4,5,7,2,1] => [1,2,7,5,4,3,6,8] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
[8,5,3,4,6,7,2,1] => [1,2,7,6,4,3,5,8] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
[8,4,3,5,6,7,2,1] => [1,2,7,6,5,3,4,8] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
[7,6,3,4,5,8,2,1] => [1,2,8,5,4,3,6,7] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
[7,5,3,4,6,8,2,1] => [1,2,8,6,4,3,5,7] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
[6,5,3,4,7,8,2,1] => [1,2,8,7,4,3,5,6] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
[6,4,3,5,7,8,2,1] => [1,2,8,7,5,3,4,6] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
[5,4,3,6,7,8,2,1] => [1,2,8,7,6,3,4,5] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
[7,8,4,3,5,2,1,6] => [6,1,2,5,3,4,8,7] => ? => ?
=> ? = 0
[8,7,4,2,1,3,5,6] => [6,5,3,1,2,4,7,8] => ? => ?
=> ? = 0
[8,6,5,3,1,2,4,7] => [7,4,2,1,3,5,6,8] => ? => ?
=> ? = 0
[8,5,4,3,1,2,6,7] => [7,6,2,1,3,4,5,8] => ? => ?
=> ? = 0
[6,5,4,2,1,3,7,8] => [8,7,3,1,2,4,5,6] => ? => ?
=> ? = 0
[4,6,7,8,5,3,2,1] => [1,2,3,5,8,7,6,4] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[4,5,7,8,6,3,2,1] => [1,2,3,6,8,7,5,4] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[4,5,6,8,7,3,2,1] => [1,2,3,7,8,6,5,4] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[3,6,7,8,5,4,2,1] => [1,2,4,5,8,7,6,3] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[3,5,7,8,6,4,2,1] => [1,2,4,6,8,7,5,3] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[3,5,6,8,7,4,2,1] => [1,2,4,7,8,6,5,3] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[3,4,7,8,6,5,2,1] => [1,2,5,6,8,7,4,3] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[3,4,6,8,7,5,2,1] => [1,2,5,7,8,6,4,3] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[3,4,5,8,7,6,2,1] => [1,2,6,7,8,5,4,3] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[6,5,4,3,8,7,2,1] => [1,2,7,8,3,4,5,6] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[2,6,7,8,5,4,3,1] => [1,3,4,5,8,7,6,2] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[2,5,7,8,6,4,3,1] => [1,3,4,6,8,7,5,2] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[2,5,6,8,7,4,3,1] => [1,3,4,7,8,6,5,2] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[2,4,7,8,6,5,3,1] => [1,3,5,6,8,7,4,2] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[2,4,6,8,7,5,3,1] => [1,3,5,7,8,6,4,2] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[2,3,7,8,6,5,4,1] => [1,4,5,6,8,7,3,2] => ? => ?
=> ? = 0
[2,3,6,8,7,5,4,1] => [1,4,5,7,8,6,3,2] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[4,3,2,6,7,8,5,1] => [1,5,8,7,6,2,3,4] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
[2,3,4,8,7,6,5,1] => [1,5,6,7,8,4,3,2] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[5,4,3,2,8,7,6,1] => [1,6,7,8,2,3,4,5] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[4,3,2,5,7,8,6,1] => [1,6,8,7,5,2,3,4] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
[4,3,2,5,6,8,7,1] => [1,7,8,6,5,2,3,4] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
[1,6,7,8,5,4,3,2] => [2,3,4,5,8,7,6,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[1,5,7,8,6,4,3,2] => [2,3,4,6,8,7,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[1,5,6,8,7,4,3,2] => [2,3,4,7,8,6,5,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[1,4,7,8,6,5,3,2] => [2,3,5,6,8,7,4,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[1,4,6,8,7,5,3,2] => [2,3,5,7,8,6,4,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[1,4,5,8,7,6,3,2] => [2,3,6,7,8,5,4,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[1,3,7,8,6,5,4,2] => [2,4,5,6,8,7,3,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
Description
The number of rises of length 2 of a Dyck path.
This is also the number of (1,1) steps of the associated Łukasiewicz path, see [1].
A related statistic is the number of double rises in a Dyck path, [[St000024]].
Matching statistic: St000884
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 57%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1,0]
=> [1] => 0
[1,2] => [2] => [1,1,0,0]
=> [1,2] => 0
[2,1] => [1,1] => [1,0,1,0]
=> [2,1] => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 0
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
[3,2,1,4,5,6,7] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 0
[4,2,1,3,5,6,7] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 0
[4,3,1,2,5,6,7] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 0
[4,3,2,1,5,6,7] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 0
[4,3,2,1,5,7,6] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[4,3,2,1,6,7,5] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[5,2,1,3,4,6,7] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 0
[5,3,1,2,4,6,7] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 0
[5,3,2,1,4,6,7] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 0
[5,3,2,1,4,7,6] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[5,3,2,1,6,7,4] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[5,4,1,2,3,6,7] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 0
[5,4,2,1,3,6,7] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 0
[5,4,2,1,3,7,6] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[5,4,2,1,6,7,3] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[5,4,3,1,2,6,7] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 0
[5,4,3,1,2,7,6] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[5,4,3,1,6,7,2] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[5,4,3,2,1,6,7] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 0
[5,4,3,2,1,7,6] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => ? = 1
[5,4,3,2,6,7,1] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[6,2,1,3,4,5,7] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 0
[6,3,1,2,4,5,7] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 0
[6,3,2,1,4,5,7] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 0
[6,3,2,1,4,7,5] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[6,3,2,1,5,7,4] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[6,4,1,2,3,5,7] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 0
[6,4,2,1,3,5,7] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 0
[6,4,2,1,3,7,5] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[6,4,2,1,5,7,3] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[6,4,3,1,2,5,7] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 0
[6,4,3,1,2,7,5] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[6,4,3,1,5,7,2] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[6,4,3,2,1,5,7] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 0
[6,4,3,2,1,7,5] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => ? = 1
[6,4,3,2,5,7,1] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[6,5,1,2,3,4,7] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 0
[6,5,2,1,3,4,7] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 0
[6,5,2,1,3,7,4] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[6,5,2,1,4,7,3] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[6,5,3,1,2,4,7] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 0
[6,5,3,1,2,7,4] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[6,5,3,1,4,7,2] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[6,5,3,2,1,4,7] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 0
[6,5,3,2,1,7,4] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => ? = 1
[6,5,3,2,4,7,1] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[6,5,4,1,2,3,7] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 0
[6,5,4,1,2,7,3] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[6,5,4,1,3,7,2] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 1
[6,5,4,2,1,3,7] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 0
Description
The number of isolated descents of a permutation.
A descent i is isolated if neither i+1 nor i−1 are descents. If a permutation has only isolated descents, then it is called primitive in [1].
Matching statistic: St001125
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001125: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 57%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001125: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[5,6,7,8,4,3,2,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[8,7,6,4,5,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,5,4,6,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,6,5,4,7,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0
[7,6,5,4,8,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,3,4,5,6,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[8,6,3,4,5,7,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[8,5,3,4,6,7,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[8,4,3,5,6,7,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[7,6,3,4,5,8,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[7,5,3,4,6,8,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[6,5,3,4,7,8,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[6,4,3,5,7,8,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[5,4,3,6,7,8,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[8,7,6,5,4,2,3,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[8,6,7,4,5,2,3,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[7,6,8,4,5,2,3,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[8,5,6,4,7,2,3,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[7,5,6,4,8,2,3,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[6,5,7,4,8,2,3,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[8,7,6,5,3,2,4,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[8,7,6,4,3,2,5,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[8,6,7,3,4,2,5,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[7,6,8,3,4,2,5,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[8,7,5,4,3,2,6,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[8,6,5,4,3,2,7,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[8,5,6,3,4,2,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[8,4,5,3,6,2,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[8,2,3,4,5,6,7,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[7,6,5,4,3,2,8,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[7,5,6,3,4,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[6,5,7,3,4,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[7,4,5,3,6,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[6,4,5,3,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[5,4,6,3,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[7,2,3,4,5,6,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[6,2,3,4,5,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[5,2,3,4,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[4,2,3,5,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[3,2,4,5,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[2,3,4,5,6,7,8,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
[7,8,6,4,5,3,1,2] => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 0
[6,7,5,4,8,3,1,2] => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 0
[7,8,5,6,3,4,1,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[6,7,5,8,3,4,1,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[5,6,7,8,3,4,1,2] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[7,8,4,5,3,6,1,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[7,8,3,4,5,6,1,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
Description
The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra.
Matching statistic: St001276
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001276: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 57%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001276: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[5,6,7,8,4,3,2,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[8,7,6,4,5,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,5,4,6,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,6,5,4,7,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0
[7,6,5,4,8,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0
[8,7,3,4,5,6,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[8,6,3,4,5,7,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[8,5,3,4,6,7,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[8,4,3,5,6,7,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[7,6,3,4,5,8,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[7,5,3,4,6,8,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[6,5,3,4,7,8,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[6,4,3,5,7,8,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[5,4,3,6,7,8,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[8,7,6,5,4,2,3,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[8,6,7,4,5,2,3,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[7,6,8,4,5,2,3,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[8,5,6,4,7,2,3,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[7,5,6,4,8,2,3,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[6,5,7,4,8,2,3,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[8,7,6,5,3,2,4,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[8,7,6,4,3,2,5,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[8,6,7,3,4,2,5,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[7,6,8,3,4,2,5,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[8,7,5,4,3,2,6,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[8,6,5,4,3,2,7,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[8,5,6,3,4,2,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[8,4,5,3,6,2,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[8,2,3,4,5,6,7,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[7,6,5,4,3,2,8,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[7,5,6,3,4,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[6,5,7,3,4,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[7,4,5,3,6,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[6,4,5,3,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[5,4,6,3,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[7,2,3,4,5,6,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[6,2,3,4,5,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[5,2,3,4,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[4,2,3,5,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[3,2,4,5,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[2,3,4,5,6,7,8,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
[7,8,6,4,5,3,1,2] => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 0
[6,7,5,4,8,3,1,2] => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 0
[7,8,5,6,3,4,1,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[6,7,5,8,3,4,1,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[5,6,7,8,3,4,1,2] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[7,8,4,5,3,6,1,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[7,8,3,4,5,6,1,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
Description
The number of 2-regular indecomposable modules in the corresponding Nakayama algebra.
Generalising the notion of k-regular modules from simple to arbitrary indecomposable modules, we call an indecomposable module M over an algebra A k-regular in case it has projective dimension k and ExtiA(M,A)=0 for i≠k and ExtkA(M,A) is 1-dimensional.
The number of Dyck paths where the statistic returns 0 might be given by [[OEIS:A035929]] .
Matching statistic: St001189
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 57%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,6,7,8,4,3,2,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,7,6,4,5,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 0
[8,7,5,4,6,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 0
[8,6,5,4,7,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 0
[7,6,5,4,8,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 0
[8,7,3,4,5,6,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[8,6,3,4,5,7,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[8,5,3,4,6,7,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[8,4,3,5,6,7,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[7,6,3,4,5,8,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[7,5,3,4,6,8,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[6,5,3,4,7,8,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[6,4,3,5,7,8,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[5,4,3,6,7,8,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[8,7,6,5,4,2,3,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1
[8,6,7,4,5,2,3,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[7,6,8,4,5,2,3,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[8,5,6,4,7,2,3,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[7,5,6,4,8,2,3,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[6,5,7,4,8,2,3,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[8,7,6,5,3,2,4,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1
[8,7,6,4,3,2,5,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1
[8,6,7,3,4,2,5,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[7,6,8,3,4,2,5,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[8,7,5,4,3,2,6,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1
[8,6,5,4,3,2,7,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1
[8,5,6,3,4,2,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[8,4,5,3,6,2,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[8,2,3,4,5,6,7,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[7,6,5,4,3,2,8,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1
[7,5,6,3,4,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[6,5,7,3,4,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[7,4,5,3,6,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[6,4,5,3,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[5,4,6,3,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[7,2,3,4,5,6,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[6,2,3,4,5,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[5,2,3,4,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[4,2,3,5,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[3,2,4,5,6,7,8,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[2,3,4,5,6,7,8,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 0
[7,8,6,4,5,3,1,2] => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 0
[6,7,5,4,8,3,1,2] => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 0
[7,8,5,6,3,4,1,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[6,7,5,8,3,4,1,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[5,6,7,8,3,4,1,2] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[7,8,4,5,3,6,1,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[7,8,3,4,5,6,1,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001274The number of indecomposable injective modules with projective dimension equal to two. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000648The number of 2-excedences of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001624The breadth of a lattice.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!