Your data matches 38 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001217
(load all 182 compositions to match this statistic)
Mapping
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001217: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => 1
[1,0,1,0]
 => [1,1,0,0]
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => 0
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 0
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 0
Description
The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1.
Matching statistic: St000745
(load all 45 compositions to match this statistic)
Mapping
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[1],[2]]
 => 2 = 1 + 1
[1,0,1,0]
 => [[1,3],[2,4]]
 => 2 = 1 + 1
[1,1,0,0]
 => [[1,2],[3,4]]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 2 = 1 + 1
[1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [[1,2,4,6],[3,5,7,8]]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [[1,2,4,5],[3,6,7,8]]
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [[1,2,3,6],[4,5,7,8]]
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [[1,2,3,5],[4,6,7,8]]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [[1,2,3,4],[5,6,7,8]]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,6,9],[2,4,7,8,10]]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,6,8],[2,4,7,9,10]]
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [[1,3,4,7,9],[2,5,6,8,10]]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [[1,3,4,7,8],[2,5,6,9,10]]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [[1,3,4,6,9],[2,5,7,8,10]]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [[1,3,4,6,8],[2,5,7,9,10]]
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [[1,2,5,7,9],[3,4,6,8,10]]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [[1,2,5,7,8],[3,4,6,9,10]]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [[1,2,5,6,9],[3,4,7,8,10]]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [[1,2,5,6,8],[3,4,7,9,10]]
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [[1,2,4,7,9],[3,5,6,8,10]]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [[1,2,4,7,8],[3,5,6,9,10]]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [[1,2,4,6,9],[3,5,7,8,10]]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [[1,2,4,6,8],[3,5,7,9,10]]
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [[1,2,4,6,7],[3,5,8,9,10]]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [[1,2,4,5,8],[3,6,7,9,10]]
 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [[1,2,4,5,7],[3,6,8,9,10]]
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [[1,2,4,5,6],[3,7,8,9,10]]
 => 1 = 0 + 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000237
Mapping
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[1],[2]]
 => [2,1] => 1
[1,0,1,0]
 => [[1,3],[2,4]]
 => [2,4,1,3] => 1
[1,1,0,0]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 0
[1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => [2,4,6,1,3,5] => 1
[1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => [2,5,6,1,3,4] => 1
[1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => [3,4,6,1,2,5] => 0
[1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => [3,5,6,1,2,4] => 0
[1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => [2,4,6,8,1,3,5,7] => 1
[1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => [2,4,7,8,1,3,5,6] => 1
[1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => [2,5,6,8,1,3,4,7] => 1
[1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => [2,5,7,8,1,3,4,6] => 1
[1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => [2,6,7,8,1,3,4,5] => 1
[1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => [3,4,6,8,1,2,5,7] => 0
[1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => [3,4,7,8,1,2,5,6] => 0
[1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => [3,5,6,8,1,2,4,7] => 0
[1,1,0,1,0,1,0,0]
 => [[1,2,4,6],[3,5,7,8]]
 => [3,5,7,8,1,2,4,6] => 0
[1,1,0,1,1,0,0,0]
 => [[1,2,4,5],[3,6,7,8]]
 => [3,6,7,8,1,2,4,5] => 0
[1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => [4,5,6,8,1,2,3,7] => 0
[1,1,1,0,0,1,0,0]
 => [[1,2,3,6],[4,5,7,8]]
 => [4,5,7,8,1,2,3,6] => 0
[1,1,1,0,1,0,0,0]
 => [[1,2,3,5],[4,6,7,8]]
 => [4,6,7,8,1,2,3,5] => 0
[1,1,1,1,0,0,0,0]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => [2,4,6,8,10,1,3,5,7,9] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => [2,4,6,9,10,1,3,5,7,8] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,6,9],[2,4,7,8,10]]
 => [2,4,7,8,10,1,3,5,6,9] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,6,8],[2,4,7,9,10]]
 => [2,4,7,9,10,1,3,5,6,8] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => [2,4,8,9,10,1,3,5,6,7] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [[1,3,4,7,9],[2,5,6,8,10]]
 => [2,5,6,8,10,1,3,4,7,9] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [[1,3,4,7,8],[2,5,6,9,10]]
 => [2,5,6,9,10,1,3,4,7,8] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [[1,3,4,6,9],[2,5,7,8,10]]
 => [2,5,7,8,10,1,3,4,6,9] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [[1,3,4,6,8],[2,5,7,9,10]]
 => [2,5,7,9,10,1,3,4,6,8] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => [2,5,8,9,10,1,3,4,6,7] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => [2,6,7,8,10,1,3,4,5,9] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => [2,6,7,9,10,1,3,4,5,8] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => [2,6,8,9,10,1,3,4,5,7] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => [2,7,8,9,10,1,3,4,5,6] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[1,2,5,7,9],[3,4,6,8,10]]
 => [3,4,6,8,10,1,2,5,7,9] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [[1,2,5,7,8],[3,4,6,9,10]]
 => [3,4,6,9,10,1,2,5,7,8] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [[1,2,5,6,9],[3,4,7,8,10]]
 => [3,4,7,8,10,1,2,5,6,9] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [[1,2,5,6,8],[3,4,7,9,10]]
 => [3,4,7,9,10,1,2,5,6,8] => 0
[1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => [3,4,8,9,10,1,2,5,6,7] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [[1,2,4,7,9],[3,5,6,8,10]]
 => [3,5,6,8,10,1,2,4,7,9] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [[1,2,4,7,8],[3,5,6,9,10]]
 => [3,5,6,9,10,1,2,4,7,8] => 0
[1,1,0,1,0,1,0,0,1,0]
 => [[1,2,4,6,9],[3,5,7,8,10]]
 => [3,5,7,8,10,1,2,4,6,9] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [[1,2,4,6,8],[3,5,7,9,10]]
 => [3,5,7,9,10,1,2,4,6,8] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [[1,2,4,6,7],[3,5,8,9,10]]
 => [3,5,8,9,10,1,2,4,6,7] => 0
[1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => [3,6,7,8,10,1,2,4,5,9] => 0
[1,1,0,1,1,0,0,1,0,0]
 => [[1,2,4,5,8],[3,6,7,9,10]]
 => [3,6,7,9,10,1,2,4,5,8] => 0
[1,1,0,1,1,0,1,0,0,0]
 => [[1,2,4,5,7],[3,6,8,9,10]]
 => [3,6,8,9,10,1,2,4,5,7] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [[1,2,4,5,6],[3,7,8,9,10]]
 => [3,7,8,9,10,1,2,4,5,6] => 0
Description
The number of small exceedances. This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St000297
(load all 14 compositions to match this statistic)
Mapping
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[1],[2]]
 => 1 => 1
[1,0,1,0]
 => [[1,3],[2,4]]
 => 101 => 1
[1,1,0,0]
 => [[1,2],[3,4]]
 => 010 => 0
[1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 10101 => 1
[1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 10010 => 1
[1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => 01001 => 0
[1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => 01010 => 0
[1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => 00100 => 0
[1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 1010101 => 1
[1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 1010010 => 1
[1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 1001001 => 1
[1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 1001010 => 1
[1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 1000100 => 1
[1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0100101 => 0
[1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => 0100010 => 0
[1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => 0101001 => 0
[1,1,0,1,0,1,0,0]
 => [[1,2,4,6],[3,5,7,8]]
 => 0101010 => 0
[1,1,0,1,1,0,0,0]
 => [[1,2,4,5],[3,6,7,8]]
 => 0100100 => 0
[1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => 0010001 => 0
[1,1,1,0,0,1,0,0]
 => [[1,2,3,6],[4,5,7,8]]
 => 0010010 => 0
[1,1,1,0,1,0,0,0]
 => [[1,2,3,5],[4,6,7,8]]
 => 0010100 => 0
[1,1,1,1,0,0,0,0]
 => [[1,2,3,4],[5,6,7,8]]
 => 0001000 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => 101010101 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => 101010010 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,6,9],[2,4,7,8,10]]
 => 101001001 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,6,8],[2,4,7,9,10]]
 => 101001010 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => 101000100 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [[1,3,4,7,9],[2,5,6,8,10]]
 => 100100101 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [[1,3,4,7,8],[2,5,6,9,10]]
 => 100100010 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [[1,3,4,6,9],[2,5,7,8,10]]
 => 100101001 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [[1,3,4,6,8],[2,5,7,9,10]]
 => 100101010 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => 100100100 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => 100010001 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => 100010010 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => 100010100 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => 100001000 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[1,2,5,7,9],[3,4,6,8,10]]
 => 010010101 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [[1,2,5,7,8],[3,4,6,9,10]]
 => 010010010 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [[1,2,5,6,9],[3,4,7,8,10]]
 => 010001001 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [[1,2,5,6,8],[3,4,7,9,10]]
 => 010001010 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => 010000100 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [[1,2,4,7,9],[3,5,6,8,10]]
 => 010100101 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [[1,2,4,7,8],[3,5,6,9,10]]
 => 010100010 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [[1,2,4,6,9],[3,5,7,8,10]]
 => 010101001 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [[1,2,4,6,8],[3,5,7,9,10]]
 => 010101010 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [[1,2,4,6,7],[3,5,8,9,10]]
 => 010100100 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => 010010001 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [[1,2,4,5,8],[3,6,7,9,10]]
 => 010010010 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [[1,2,4,5,7],[3,6,8,9,10]]
 => 010010100 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [[1,2,4,5,6],[3,7,8,9,10]]
 => 010001000 => 0
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 8 compositions to match this statistic)
Mapping
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 10 => 01 => 2 = 1 + 1
[1,0,1,0]
 => 1010 => 0101 => 2 = 1 + 1
[1,1,0,0]
 => 1100 => 1001 => 1 = 0 + 1
[1,0,1,0,1,0]
 => 101010 => 010101 => 2 = 1 + 1
[1,0,1,1,0,0]
 => 101100 => 011001 => 2 = 1 + 1
[1,1,0,0,1,0]
 => 110010 => 100101 => 1 = 0 + 1
[1,1,0,1,0,0]
 => 110100 => 101001 => 1 = 0 + 1
[1,1,1,0,0,0]
 => 111000 => 110001 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => 10101010 => 01010101 => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
 => 10101100 => 01011001 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => 10110010 => 01100101 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => 10110100 => 01101001 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => 10111000 => 01110001 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => 11001010 => 10010101 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => 11001100 => 10011001 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => 11010010 => 10100101 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => 11010100 => 10101001 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => 11011000 => 10110001 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => 11100010 => 11000101 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => 11100100 => 11001001 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => 11101000 => 11010001 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => 11110000 => 11100001 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 0101010101 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 0101011001 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 0101100101 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 0101101001 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 0101110001 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 0110010101 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 0110011001 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 0110100101 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 0110101001 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 0110110001 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 0111000101 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 0111001001 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 0111010001 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 0111100001 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 1001010101 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 1001011001 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 1001100101 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 1001101001 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 1001110001 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 1010010101 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 1010011001 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 1010100101 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => 1010101001 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => 1010110001 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 1011000101 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => 1011001001 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => 1011010001 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 1011100001 => 1 = 0 + 1
Description
The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St001594
(load all 2 compositions to match this statistic)
Mapping
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001594: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [.,.]
 => [1,0]
 => 1
[1,0,1,0]
 => [2,1] => [[.,.],.]
 => [1,1,0,0]
 => 1
[1,1,0,0]
 => [1,2] => [.,[.,.]]
 => [1,0,1,0]
 => 0
[1,0,1,0,1,0]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 1
[1,0,1,1,0,0]
 => [2,3,1] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,0,1,0]
 => [3,1,2] => [[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => 0
[1,1,0,1,0,0]
 => [1,3,2] => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 0
[1,1,1,0,0,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
Description
The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. See the link for the definition.
Matching statistic: St001204
(load all 176 compositions to match this statistic)
Mapping
St001204: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => ? = 1
[1,0,1,0]
 => 1
[1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => 1
[1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => 0
[1,1,0,1,0,0]
 => 0
[1,1,1,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,0]
 => 0
[1,1,1,0,0,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => 0
[1,1,1,0,1,0,0,0]
 => 0
[1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => 0
[1,1,1,0,0,0,1,0,1,0]
 => 0
Description
Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. Associate to this special CNakayama algebra a Dyck path as follows: In the list L delete the first entry $c_0$ and substract from all other entries $n$−1 and then append the last element 1. The result is a Kupisch series of an LNakayama algebra. The statistic gives the $(t-1)/2$ when $t$ is the projective dimension of the simple module $S_{n-2}$.
Matching statistic: St000990
(load all 69 compositions to match this statistic)
Mapping
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000990: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => ? = 1 + 1
[1,0,1,0]
 => [2,1] => [2,1] => 2 = 1 + 1
[1,1,0,0]
 => [1,2] => [1,2] => 1 = 0 + 1
[1,0,1,0,1,0]
 => [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [3,1,2] => 2 = 1 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [2,3,1] => 1 = 0 + 1
[1,1,0,1,0,0]
 => [1,3,2] => [1,3,2] => 1 = 0 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [3,1,4,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [3,1,2,4] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [2,4,1,3] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,4,1,2] => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [2,3,1,4] => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [1,3,2,4] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [1,4,2,3] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,3,4,2] => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [1,2,4,3] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [3,1,4,2,5] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [2,1,5,3,4] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [3,1,5,2,4] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [4,1,5,2,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [2,1,4,5,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [3,1,4,5,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [3,1,2,5,4] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,1,2,5,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [2,4,1,3,5] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [3,4,1,2,5] => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [2,5,1,3,4] => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [3,5,1,2,4] => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [4,5,1,2,3] => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [2,4,1,5,3] => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [3,4,1,5,2] => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [2,3,1,5,4] => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,4,2,5,3] => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [2,3,1,4,5] => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => [1,4,2,3,5] => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => [1,5,2,3,4] => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [2,4,5,1,3] => 1 = 0 + 1
Description
The first ascent of a permutation. For a permutation $\pi$, this is the smallest index such that $\pi(i) < \pi(i+1)$. For the first descent, see [[St000654]].
Matching statistic: St000390
(load all 4 compositions to match this statistic)
Mapping
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => => ? = 1
[1,0,1,0]
 => [2,1] => [1,2] => 1 => 1
[1,1,0,0]
 => [1,2] => [2,1] => 0 => 0
[1,0,1,0,1,0]
 => [2,1,3] => [1,3,2] => 10 => 1
[1,0,1,1,0,0]
 => [2,3,1] => [1,2,3] => 11 => 1
[1,1,0,0,1,0]
 => [3,1,2] => [3,1,2] => 00 => 0
[1,1,0,1,0,0]
 => [1,3,2] => [3,2,1] => 00 => 0
[1,1,1,0,0,0]
 => [1,2,3] => [2,3,1] => 00 => 0
[1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [1,4,3,2] => 100 => 1
[1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [1,4,2,3] => 100 => 1
[1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [1,3,4,2] => 100 => 1
[1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [1,2,4,3] => 110 => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => 111 => 1
[1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [4,1,3,2] => 000 => 0
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [4,1,2,3] => 000 => 0
[1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [3,1,4,2] => 000 => 0
[1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [3,2,4,1] => 000 => 0
[1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [4,2,3,1] => 000 => 0
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [3,4,1,2] => 000 => 0
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [3,4,2,1] => 000 => 0
[1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [2,4,3,1] => 000 => 0
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [2,3,4,1] => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [1,4,3,5,2] => 1000 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [1,4,2,5,3] => 1000 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [1,5,3,4,2] => 1000 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [1,5,2,4,3] => 1000 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [1,5,2,3,4] => 1000 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [1,4,5,3,2] => 1000 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [1,4,5,2,3] => 1000 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [1,3,5,4,2] => 1000 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [1,2,5,4,3] => 1100 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [1,2,5,3,4] => 1100 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [1,3,4,5,2] => 1000 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [1,2,4,5,3] => 1100 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [1,2,3,5,4] => 1110 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 1111 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [4,1,3,5,2] => 0000 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [4,1,2,5,3] => 0000 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [5,1,3,4,2] => 0000 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [5,1,2,4,3] => 0000 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [5,1,2,3,4] => 0000 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [4,1,5,3,2] => 0000 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [4,1,5,2,3] => 0000 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [3,1,5,4,2] => 0000 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [3,2,5,4,1] => 0000 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [4,2,5,3,1] => 0000 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [3,1,4,5,2] => 0000 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => [3,2,4,5,1] => 0000 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => [4,2,3,5,1] => 0000 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => [5,2,3,4,1] => 0000 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [4,5,1,3,2] => 0000 => 0
Description
The number of runs of ones in a binary word.
Matching statistic: St000541
(load all 3 compositions to match this statistic)
Mapping
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [[1]]
 => [1] => ? = 1
[1,0,1,0]
 => [1,1,0,0]
 => [[0,1],[1,0]]
 => [2,1] => 1
[1,1,0,0]
 => [1,0,1,0]
 => [[1,0],[0,1]]
 => [1,2] => 0
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [3,1,2] => 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [2,1,3] => 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,3,2] => 0
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [1,2,3] => 0
[1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,3,2] => 0
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [3,1,2,4] => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [2,1,4,3] => 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [2,1,3,4] => 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [2,1,4,3] => 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 0
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 0
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,2,4,3] => 0
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,2,3,4] => 0
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 0
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 0
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 0
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 0
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [5,1,2,3,4] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [4,1,2,3,5] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [3,1,2,5,4] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [3,1,2,4,5] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [3,1,2,5,4] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [2,1,5,3,4] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [2,1,4,3,5] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [2,1,3,4,5] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [2,1,5,3,4] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [2,1,4,3,5] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [2,1,5,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,5,2,3,4] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => 0
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 0
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,2,3,4,5] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 0
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 0
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 0
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 0
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,5,2,3,4] => 0
Description
The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
The following 28 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000989The number of final rises of a permutation. St000654The first descent of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000264The girth of a graph, which is not a tree. St000877The depth of the binary word interpreted as a path. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000456The monochromatic index of a connected graph. St001948The number of augmented double ascents of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000382The first part of an integer composition. St001545The second Elser number of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000392The length of the longest run of ones in a binary word. St000383The last part of an integer composition. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid.