- FindStat

Your data matches 40 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001217
(load all 109 compositions to match this statistic)

Mapping

St001217: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => 1
[1,0,1,0]
 => 0
[1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => 0
[1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,0]
 => 0
[1,0,1,1,1,0,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => 1
[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]
 => 1
[1,1,1,0,0,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => 0
[1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => 1
[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

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 23 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [2,1] => [[1],[2]]
 => 2 = 1 + 1
[1,0,1,0]
 => [3,1,2] => [[1,2],[3]]
 => 1 = 0 + 1
[1,1,0,0]
 => [2,3,1] => [[1,3],[2]]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [[1,2,3],[4]]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [[1,2],[3,4]]
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [[1,3],[2,4]]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [[1,2],[3],[4]]
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [[1,3,4],[2]]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [[1,2,3,4],[5]]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [[1,2,3],[4,5]]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [[1,2,4],[3,5]]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [[1,2,3],[4],[5]]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [[1,2,5],[3,4]]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [[1,3,4],[2,5]]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [[1,3,5],[2,4]]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [[1,2,4],[3],[5]]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [[1,2,3],[4],[5]]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [[1,2],[3,5],[4]]
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [[1,3,4],[2,5]]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [[1,3],[2,4],[5]]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [[1,2],[3,4],[5]]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [[1,3,4,5],[2]]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [[1,2,3,4,5],[6]]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [[1,2,3,4],[5,6]]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [[1,2,3,5],[4,6]]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [[1,2,3,4],[5],[6]]
 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [[1,2,3,6],[4,5]]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [[1,2,4,5],[3,6]]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [[1,2,4],[3,5,6]]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [[1,2,3,5],[4],[6]]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [[1,2,3,4],[5],[6]]
 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [[1,2,3],[4,6],[5]]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [[1,2,5],[3,4,6]]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [[1,2,4],[3,5],[6]]
 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [[1,2,3],[4,5],[6]]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [[1,2,5,6],[3,4]]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [[1,3,4,5],[2,6]]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [[1,3,4],[2,5,6]]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [[1,3,5],[2,4,6]]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [[1,3,4],[2,5],[6]]
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [[1,3,5,6],[2,4]]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [[1,2,4,5],[3],[6]]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [[1,2,4],[3,6],[5]]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [[1,2,3,5],[4],[6]]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [[1,2,3,4],[5,6]]
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [[1,2,3],[4,6],[5]]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [[1,2,5],[3,6],[4]]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [[1,2,4],[3,5],[6]]
 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [[1,2,3],[4,5],[6]]
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [[1,2,6],[3,5],[4]]
 => 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
(load all 3 compositions to match this statistic)

Mapping

Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00226: Standard tableaux —row-to-column-descents⟶ 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]]
 => [[1],[2]]
 => [2,1] => 1
[1,0,1,0]
 => [[1,3],[2,4]]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 0
[1,1,0,0]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => [2,4,1,3] => 1
[1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => [[1,2,4],[3,5,6]]
 => [3,5,6,1,2,4] => 0
[1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => [[1,2,5],[3,4,6]]
 => [3,4,6,1,2,5] => 0
[1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => [[1,3,4],[2,5,6]]
 => [2,5,6,1,3,4] => 1
[1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => 0
[1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => [[1,3,5],[2,4,6]]
 => [2,4,6,1,3,5] => 1
[1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => [[1,2,4,6],[3,5,7,8]]
 => [3,5,7,8,1,2,4,6] => 0
[1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => [[1,2,4,7],[3,5,6,8]]
 => [3,5,6,8,1,2,4,7] => 0
[1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => [[1,2,5,6],[3,4,7,8]]
 => [3,4,7,8,1,2,5,6] => 0
[1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => [[1,2,4,5],[3,6,7,8]]
 => [3,6,7,8,1,2,4,5] => 0
[1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => [[1,2,5,7],[3,4,6,8]]
 => [3,4,6,8,1,2,5,7] => 0
[1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => [[1,3,4,6],[2,5,7,8]]
 => [2,5,7,8,1,3,4,6] => 1
[1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => [[1,3,4,7],[2,5,6,8]]
 => [2,5,6,8,1,3,4,7] => 1
[1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => [[1,2,3,6],[4,5,7,8]]
 => [4,5,7,8,1,2,3,6] => 0
[1,1,0,1,0,1,0,0]
 => [[1,2,4,6],[3,5,7,8]]
 => [[1,2,3,5],[4,6,7,8]]
 => [4,6,7,8,1,2,3,5] => 0
[1,1,0,1,1,0,0,0]
 => [[1,2,4,5],[3,6,7,8]]
 => [[1,2,3,7],[4,5,6,8]]
 => [4,5,6,8,1,2,3,7] => 0
[1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => [[1,3,5,6],[2,4,7,8]]
 => [2,4,7,8,1,3,5,6] => 1
[1,1,1,0,0,1,0,0]
 => [[1,2,3,6],[4,5,7,8]]
 => [[1,3,4,5],[2,6,7,8]]
 => [2,6,7,8,1,3,4,5] => 1
[1,1,1,0,1,0,0,0]
 => [[1,2,3,5],[4,6,7,8]]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => 0
[1,1,1,1,0,0,0,0]
 => [[1,2,3,4],[5,6,7,8]]
 => [[1,3,5,7],[2,4,6,8]]
 => [2,4,6,8,1,3,5,7] => 1
[1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => [[1,2,4,6,8],[3,5,7,9,10]]
 => [3,5,7,9,10,1,2,4,6,8] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => [[1,2,4,6,9],[3,5,7,8,10]]
 => [3,5,7,8,10,1,2,4,6,9] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,6,9],[2,4,7,8,10]]
 => [[1,2,4,7,8],[3,5,6,9,10]]
 => [3,5,6,9,10,1,2,4,7,8] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,6,8],[2,4,7,9,10]]
 => [[1,2,4,6,7],[3,5,8,9,10]]
 => [3,5,8,9,10,1,2,4,6,7] => 0
[1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => [[1,2,4,7,9],[3,5,6,8,10]]
 => [3,5,6,8,10,1,2,4,7,9] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [[1,3,4,7,9],[2,5,6,8,10]]
 => [[1,2,5,6,8],[3,4,7,9,10]]
 => [3,4,7,9,10,1,2,5,6,8] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [[1,3,4,7,8],[2,5,6,9,10]]
 => [[1,2,5,6,9],[3,4,7,8,10]]
 => [3,4,7,8,10,1,2,5,6,9] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [[1,3,4,6,9],[2,5,7,8,10]]
 => [[1,2,4,5,8],[3,6,7,9,10]]
 => [3,6,7,9,10,1,2,4,5,8] => 0
[1,0,1,1,0,1,0,1,0,0]
 => [[1,3,4,6,8],[2,5,7,9,10]]
 => [[1,2,4,5,7],[3,6,8,9,10]]
 => [3,6,8,9,10,1,2,4,5,7] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => [3,6,7,8,10,1,2,4,5,9] => 0
[1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => [[1,2,5,7,8],[3,4,6,9,10]]
 => [3,4,6,9,10,1,2,5,7,8] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => [3,4,8,9,10,1,2,5,6,7] => 0
[1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => [[1,2,4,5,6],[3,7,8,9,10]]
 => [3,7,8,9,10,1,2,4,5,6] => 0
[1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => [[1,2,5,7,9],[3,4,6,8,10]]
 => [3,4,6,8,10,1,2,5,7,9] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [[1,2,5,7,9],[3,4,6,8,10]]
 => [[1,3,4,6,8],[2,5,7,9,10]]
 => [2,5,7,9,10,1,3,4,6,8] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [[1,2,5,7,8],[3,4,6,9,10]]
 => [[1,3,4,6,9],[2,5,7,8,10]]
 => [2,5,7,8,10,1,3,4,6,9] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [[1,2,5,6,9],[3,4,7,8,10]]
 => [[1,3,4,7,8],[2,5,6,9,10]]
 => [2,5,6,9,10,1,3,4,7,8] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [[1,2,5,6,8],[3,4,7,9,10]]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => [2,5,8,9,10,1,3,4,6,7] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => [[1,3,4,7,9],[2,5,6,8,10]]
 => [2,5,6,8,10,1,3,4,7,9] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [[1,2,4,7,9],[3,5,6,8,10]]
 => [[1,2,3,6,8],[4,5,7,9,10]]
 => [4,5,7,9,10,1,2,3,6,8] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [[1,2,4,7,8],[3,5,6,9,10]]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => [4,5,7,8,10,1,2,3,6,9] => 0
[1,1,0,1,0,1,0,0,1,0]
 => [[1,2,4,6,9],[3,5,7,8,10]]
 => [[1,2,3,5,8],[4,6,7,9,10]]
 => [4,6,7,9,10,1,2,3,5,8] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [[1,2,4,6,8],[3,5,7,9,10]]
 => [[1,2,3,5,7],[4,6,8,9,10]]
 => [4,6,8,9,10,1,2,3,5,7] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [[1,2,4,6,7],[3,5,8,9,10]]
 => [[1,2,3,5,9],[4,6,7,8,10]]
 => [4,6,7,8,10,1,2,3,5,9] => 0
[1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => [4,5,6,9,10,1,2,3,7,8] => 0
[1,1,0,1,1,0,0,1,0,0]
 => [[1,2,4,5,8],[3,6,7,9,10]]
 => [[1,2,3,6,7],[4,5,8,9,10]]
 => [4,5,8,9,10,1,2,3,6,7] => 0
[1,1,0,1,1,0,1,0,0,0]
 => [[1,2,4,5,7],[3,6,8,9,10]]
 => [[1,2,3,5,6],[4,7,8,9,10]]
 => [4,7,8,9,10,1,2,3,5,6] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [[1,2,4,5,6],[3,7,8,9,10]]
 => [[1,2,3,7,9],[4,5,6,8,10]]
 => [4,5,6,8,10,1,2,3,7,9] => 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 9 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00109: Permutations —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]
 => [2,1] => [2,1] => 1 => 1
[1,0,1,0]
 => [3,1,2] => [2,3,1] => 01 => 0
[1,1,0,0]
 => [2,3,1] => [3,1,2] => 10 => 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => 001 => 0
[1,0,1,1,0,0]
 => [3,1,4,2] => [2,4,1,3] => 010 => 0
[1,1,0,0,1,0]
 => [2,4,1,3] => [3,1,4,2] => 101 => 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [3,4,2,1] => 011 => 0
[1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => 100 => 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [2,3,4,5,1] => 0001 => 0
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [2,3,5,1,4] => 0010 => 0
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [2,4,1,5,3] => 0101 => 0
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [2,4,5,3,1] => 0011 => 0
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [2,5,1,3,4] => 0100 => 0
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [3,1,4,5,2] => 1001 => 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [3,1,5,2,4] => 1010 => 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,4,2,5,1] => 0101 => 0
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [3,4,5,2,1] => 0011 => 0
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [3,5,2,1,4] => 0110 => 0
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [4,1,2,5,3] => 1001 => 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [4,1,5,3,2] => 1011 => 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [4,5,2,3,1] => 0101 => 0
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => 1000 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [2,3,4,5,6,1] => 00001 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [2,3,4,6,1,5] => 00010 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [2,3,5,1,6,4] => 00101 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [2,3,5,6,4,1] => 00011 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [2,3,6,1,4,5] => 00100 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [2,4,1,5,6,3] => 01001 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [2,4,1,6,3,5] => 01010 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [2,4,5,3,6,1] => 00101 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [2,4,5,6,3,1] => 00011 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [2,4,6,3,1,5] => 00110 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [2,5,1,3,6,4] => 01001 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [2,5,1,6,4,3] => 01011 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [2,5,6,3,4,1] => 00101 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [2,6,1,3,4,5] => 01000 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [3,1,4,5,6,2] => 10001 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [3,1,4,6,2,5] => 10010 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [3,1,5,2,6,4] => 10101 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [3,1,5,6,4,2] => 10011 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [3,1,6,2,4,5] => 10100 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [3,4,2,5,6,1] => 01001 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [3,4,2,6,1,5] => 01010 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [3,4,5,2,6,1] => 00101 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [3,4,5,6,1,2] => 00010 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [3,4,6,2,1,5] => 00110 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [3,5,2,1,6,4] => 01101 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [3,5,2,6,4,1] => 01011 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [3,5,6,2,4,1] => 00101 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [3,6,2,1,4,5] => 01100 => 0

Description

The number of leading ones in a binary word.

Matching statistic: St000390

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [2,1] => [1,2] => 1 => 1
[1,0,1,0]
 => [3,1,2] => [3,1,2] => 00 => 0
[1,1,0,0]
 => [2,3,1] => [1,2,3] => 11 => 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [3,4,1,2] => 000 => 0
[1,0,1,1,0,0]
 => [3,1,4,2] => [4,1,3,2] => 000 => 0
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,4,2,3] => 100 => 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [4,2,1,3] => 000 => 0
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => 111 => 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [3,4,5,1,2] => 0000 => 0
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [3,5,1,4,2] => 0000 => 0
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [4,1,5,3,2] => 0000 => 0
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [4,5,3,1,2] => 0000 => 0
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [5,1,3,4,2] => 0000 => 0
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,4,5,2,3] => 1000 => 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,5,2,4,3] => 1000 => 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [4,2,5,1,3] => 0000 => 0
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [4,5,2,1,3] => 0000 => 0
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [5,2,1,4,3] => 0000 => 0
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,2,5,3,4] => 1100 => 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,5,3,2,4] => 1000 => 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [5,2,3,1,4] => 0000 => 0
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 1111 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [3,4,5,6,1,2] => 00000 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [3,4,6,1,5,2] => 00000 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [3,5,1,6,4,2] => 00000 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [3,5,6,4,1,2] => 00000 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [3,6,1,4,5,2] => 00000 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [4,1,5,6,3,2] => 00000 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [4,1,6,3,5,2] => 00000 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [4,5,3,6,1,2] => 00000 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [4,5,6,3,1,2] => 00000 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [4,6,3,1,5,2] => 00000 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [5,1,3,6,4,2] => 00000 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [5,1,6,4,3,2] => 00000 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [5,6,3,4,1,2] => 00000 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [6,1,3,4,5,2] => 00000 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,4,5,6,2,3] => 10000 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,4,6,2,5,3] => 10000 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,5,2,6,4,3] => 10000 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,5,6,4,2,3] => 10000 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,6,2,4,5,3] => 10000 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [4,2,5,6,1,3] => 00000 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [4,2,6,1,5,3] => 00000 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,5,2,6,1,3] => 00000 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [4,5,6,1,2,3] => 00000 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,6,2,1,5,3] => 00000 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [5,2,1,6,4,3] => 00000 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [5,2,6,4,1,3] => 00000 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [5,6,2,4,1,3] => 00000 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [6,2,1,4,5,3] => 00000 => 0

Description

The number of runs of ones in a binary word.

Matching statistic: St001594

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ 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]
 => [1,2] => [.,[.,.]]
 => [1,0,1,0]
 => 0
[1,1,0,0]
 => [2,1] => [[.,.],.]
 => [1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => [1,3,2] => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => 0
[1,1,1,0,0,0]
 => [3,1,2] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 0
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 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: St000326
(load all 6 compositions to match this statistic)

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
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]
 => [1,0]
 => 10 => 01 => 2 = 1 + 1
[1,0,1,0]
 => [1,1,0,0]
 => 1100 => 1001 => 1 = 0 + 1
[1,1,0,0]
 => [1,0,1,0]
 => 1010 => 0101 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 111000 => 110001 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 110010 => 100101 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 101100 => 011001 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 110100 => 101001 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 101010 => 010101 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 11100001 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 11000101 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 10011001 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 11001001 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 10010101 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 01110001 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 01100101 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 11011000 => 10110001 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => 11010001 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 10100101 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 01011001 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 01101001 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 10101001 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 01010101 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 1111000001 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 1110000101 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 1100011001 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => 1110001001 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 1100010101 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 1001110001 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 1001100101 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => 1100110001 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => 1110010001 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 1100100101 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 1001011001 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 1001101001 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => 1100101001 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 1001010101 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 0111100001 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 0111000101 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 0110011001 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 0111001001 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 0110010101 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 1011100001 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 1011000101 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => 1101100001 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => 1110100001 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => 1101000101 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 1010011001 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [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]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => 1101001001 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 1010010101 => 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: St000678

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [2,1] => [1,2] => [1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0]
 => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0]
 => [2,3,1] => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [3,6,5,1,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,6,5,2,4,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [3,6,5,2,1,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [4,6,1,5,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [4,6,2,5,1,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,6,3,5,4,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,6,2,5,4,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [2,6,3,5,1,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [4,6,3,1,5,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [4,6,1,2,5,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [4,6,3,2,1,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [5,1,6,4,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [5,3,6,1,4,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [5,1,6,2,4,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [5,3,6,2,1,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,4,6,5,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [2,4,6,5,1,3] => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,3,6,5,4,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [2,3,6,5,1,4] => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [3,4,6,1,5,2] => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [3,4,6,2,1,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1

Description

The number of up steps after the last double rise of a Dyck path.

Matching statistic: St001135

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001135: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [2,1] => [1,2] => [1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0]
 => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0]
 => [2,3,1] => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [3,6,5,1,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,6,5,2,4,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [3,6,5,2,1,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [4,6,1,5,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [4,6,2,5,1,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,6,3,5,4,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,6,2,5,4,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [2,6,3,5,1,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [4,6,3,1,5,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [4,6,1,2,5,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [4,6,3,2,1,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [5,1,6,4,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [5,3,6,1,4,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [5,1,6,2,4,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [5,3,6,2,1,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,4,6,5,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [2,4,6,5,1,3] => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,3,6,5,4,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [2,3,6,5,1,4] => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [3,4,6,1,5,2] => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [3,4,6,2,1,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1

Description

The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001204
(load all 163 compositions to match this statistic)

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001204: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => ? = 1
[1,0,1,0]
 => [1,1,0,0]
 => 0
[1,1,0,0]
 => [1,0,1,0]
 => 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 0
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 0
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1

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

$S_{n-2}$ .

The following 30 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000990The first ascent of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000989The number of final rises of a permutation. St000654The first descent of a permutation. St000877The depth of the binary word interpreted as a path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000061The number of nodes on the left branch of a binary tree. St000504The cardinality of the first block of a set partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000352The Elizalde-Pak rank of a permutation. St000234The number of global ascents of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000542The number of left-to-right-minima of a permutation. 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

$eA$ . St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. 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

$Af$ . St001545The second Elser number of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. 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

$eA$ . 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.