- FindStat

Your data matches 166 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000025
(load all 36 compositions to match this statistic)

Mapping

St000025: 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]
 => 1
[1,1,0,0]
 => 2
[1,0,1,0,1,0]
 => 1
[1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0]
 => 3
[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]
 => 2
[1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => 2
[1,1,0,1,0,1,0,0]
 => 2
[1,1,0,1,1,0,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => 3
[1,1,1,0,0,1,0,0]
 => 3
[1,1,1,0,1,0,0,0]
 => 3
[1,1,1,1,0,0,0,0]
 => 4
[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]
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => 2

Description

The number of initial rises of a Dyck path. In other words, this is the height of the first peak of

$D$ .

Matching statistic: St000011
(load all 43 compositions to match this statistic)

Mapping

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: 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]
 => 2
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[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]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[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]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

Matching statistic: St000745
(load all 12 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ 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]
 => [1] => [[1]]
 => 1
[1,0,1,0]
 => [1,2] => [[1,2]]
 => 1
[1,1,0,0]
 => [2,1] => [[1],[2]]
 => 2
[1,0,1,0,1,0]
 => [1,2,3] => [[1,2,3]]
 => 1
[1,0,1,1,0,0]
 => [1,3,2] => [[1,2],[3]]
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [[1,3],[2]]
 => 2
[1,1,0,1,0,0]
 => [2,3,1] => [[1,3],[2]]
 => 2
[1,1,1,0,0,0]
 => [3,2,1] => [[1],[2],[3]]
 => 3
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [[1,2,3,4]]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [[1,2,3],[4]]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [[1,2,4],[3]]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [[1,2,4],[3]]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [[1,2],[3],[4]]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [[1,3,4],[2]]
 => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [[1,3],[2,4]]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [[1,3,4],[2]]
 => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [[1,3,4],[2]]
 => 2
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [[1,3],[2],[4]]
 => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [[1,4],[2],[3]]
 => 3
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [[1,4],[2],[3]]
 => 3
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [[1,4],[2],[3]]
 => 3
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [[1],[2],[3],[4]]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [[1,2,3,4,5]]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [[1,2,3,5],[4]]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [[1,2,3,5],[4]]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [[1,2,3],[4],[5]]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [[1,2,4,5],[3]]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [[1,2,4,5],[3]]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [[1,2,4,5],[3]]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [[1,2,4],[3],[5]]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [[1,2,5],[3],[4]]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [[1,2,5],[3],[4]]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [[1,2,5],[3],[4]]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [[1,3,4],[2,5]]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [[1,3,5],[2,4]]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [[1,3,5],[2,4]]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [[1,3],[2,4],[5]]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [[1,3,4,5],[2]]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [[1,3,4],[2,5]]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [[1,3,4,5],[2]]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [[1,3,4,5],[2]]
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [[1,3,4],[2],[5]]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [[1,3,5],[2],[4]]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [[1,3,5],[2],[4]]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [[1,3,5],[2],[4]]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [[1,3],[2],[4],[5]]
 => 2

Description

The index of the last row whose first entry is the row number in a standard Young tableau.

Matching statistic: St000382
(load all 19 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The first part of an integer composition.

Matching statistic: St000439
(load all 40 compositions to match this statistic)

Mapping

St000439: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 100%●distinct values known / distinct values provided: 88%

Values

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

Description

The position of the first down step of a Dyck path.

Matching statistic: St000971
(load all 30 compositions to match this statistic)

Mapping

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 88% ●values known / values provided: 100%●distinct values known / distinct values provided: 88%

Values

[1,0]
 => {{1}}
 => 1
[1,0,1,0]
 => {{1},{2}}
 => 1
[1,1,0,0]
 => {{1,2}}
 => 2
[1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 1
[1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1
[1,1,0,0,1,0]
 => {{1,2},{3}}
 => 2
[1,1,0,1,0,0]
 => {{1,3},{2}}
 => 2
[1,1,1,0,0,0]
 => {{1,2,3}}
 => 3
[1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 1
[1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 1
[1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1
[1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 1
[1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 1
[1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 2
[1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2
[1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 2
[1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 2
[1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 2
[1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 3
[1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 3
[1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 3
[1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => {{1,3,5},{2},{4}}
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => 2
[]
 => {}
 => ? = 0

Description

The smallest closer of a set partition. A closer (or right hand endpoint) of a set partition is a number that is maximal in its block. For this statistic, singletons are considered as closers. In other words, this is the smallest among the maximal elements of the blocks.

Matching statistic: St000026
(load all 5 compositions to match this statistic)

Mapping

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 100%●distinct values known / distinct values provided: 88%

Values

[1,0]
 => [1] => [1,0]
 => 1
[1,0,1,0]
 => [1,1] => [1,0,1,0]
 => 1
[1,1,0,0]
 => [2] => [1,1,0,0]
 => 2
[1,0,1,0,1,0]
 => [1,1,1] => [1,0,1,0,1,0]
 => 1
[1,0,1,1,0,0]
 => [1,2] => [1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => [2,1] => [1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => [2,1] => [1,1,0,0,1,0]
 => 2
[1,1,1,0,0,0]
 => [3] => [1,1,1,0,0,0]
 => 3
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,1,0,1,1,0,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[1,1,1,0,0,1,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[1,1,1,0,1,0,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1,0,0,0,0]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[]
 => [] => ?
 => ? = 0

Description

The position of the first return of a Dyck path.

Matching statistic: St000383
(load all 17 compositions to match this statistic)

Mapping

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 88% ●values known / values provided: 100%●distinct values known / distinct values provided: 88%

Values

[1,0]
 => [1] => [1] => 1
[1,0,1,0]
 => [1,1] => [1,1] => 1
[1,1,0,0]
 => [2] => [2] => 2
[1,0,1,0,1,0]
 => [1,1,1] => [1,1,1] => 1
[1,0,1,1,0,0]
 => [1,2] => [2,1] => 1
[1,1,0,0,1,0]
 => [2,1] => [1,2] => 2
[1,1,0,1,0,0]
 => [2,1] => [1,2] => 2
[1,1,1,0,0,0]
 => [3] => [3] => 3
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1] => 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1,1] => 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [1,2,1] => 1
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [1,2,1] => 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [3,1] => 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,1,2] => 2
[1,1,0,0,1,1,0,0]
 => [2,2] => [2,2] => 2
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [1,1,2] => 2
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,1,2] => 2
[1,1,0,1,1,0,0,0]
 => [2,2] => [2,2] => 2
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,3] => 3
[1,1,1,0,0,1,0,0]
 => [3,1] => [1,3] => 3
[1,1,1,0,1,0,0,0]
 => [3,1] => [1,3] => 3
[1,1,1,1,0,0,0,0]
 => [4] => [4] => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,1,1,1,1] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [2,1,1,1] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [1,2,1,1] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [1,2,1,1] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [3,1,1] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [1,1,2,1] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [2,2,1] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [1,1,2,1] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [1,1,2,1] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [2,2,1] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [1,3,1] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [1,3,1] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [1,3,1] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [4,1] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,1,1,2] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [2,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [1,2,2] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [1,2,2] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [3,2] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [1,1,1,2] => 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [2,1,2] => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [1,1,1,2] => 2
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,1,1,2] => 2
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [2,1,2] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [1,2,2] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [1,2,2] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [1,2,2] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [3,2] => 2
[]
 => [] => [] => ? = 0

Description

The last part of an integer composition.

Matching statistic: St001784
(load all 7 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St001784: Set partitions ⟶ ℤResult quality: 88% ●values known / values provided: 100%●distinct values known / distinct values provided: 88%

Values

[1,0]
 => [1] => {{1}}
 => 1
[1,0,1,0]
 => [1,2] => {{1},{2}}
 => 1
[1,1,0,0]
 => [2,1] => {{1,2}}
 => 2
[1,0,1,0,1,0]
 => [1,2,3] => {{1},{2},{3}}
 => 1
[1,0,1,1,0,0]
 => [1,3,2] => {{1},{2,3}}
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => {{1,2},{3}}
 => 2
[1,1,0,1,0,0]
 => [2,3,1] => {{1,2,3}}
 => 2
[1,1,1,0,0,0]
 => [3,1,2] => {{1,3},{2}}
 => 3
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => {{1},{2},{3},{4}}
 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => {{1},{2},{3,4}}
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => {{1},{2,3},{4}}
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => {{1},{2,3,4}}
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => {{1},{2,4},{3}}
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => {{1,2},{3},{4}}
 => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => {{1,2},{3,4}}
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => {{1,2,3},{4}}
 => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => {{1,2,3,4}}
 => 2
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => {{1,2,4},{3}}
 => 2
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => {{1,3},{2},{4}}
 => 3
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => {{1,3,4},{2}}
 => 3
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => {{1,3},{2,4}}
 => 3
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => {{1,4},{2},{3}}
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => {{1},{2,3,4},{5}}
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => {{1},{2,3,4,5}}
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => {{1},{2,3,5},{4}}
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => {{1},{2,4},{3},{5}}
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => {{1},{2,4,5},{3}}
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => {{1},{2,4},{3,5}}
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => {{1},{2,5},{3},{4}}
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => {{1,2},{3},{4,5}}
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => {{1,2},{3,4},{5}}
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => {{1,2},{3,4,5}}
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => {{1,2},{3,5},{4}}
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => {{1,2,3},{4},{5}}
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => {{1,2,3},{4,5}}
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => {{1,2,3,4},{5}}
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => {{1,2,3,5},{4}}
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => {{1,2,4},{3},{5}}
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => {{1,2,4,5},{3}}
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => {{1,2,4},{3,5}}
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => {{1,2,5},{3},{4}}
 => 2
[]
 => [] => ?
 => ? = 0

Description

The minimum of the smallest closer and the second element of the block containing 1 in a set partition. A closer of a set partition is the maximal element of a non-singleton block. This statistic is defined as

$1$ if

$\{1\}$ is a singleton block, and otherwise the minimum of the smallest closer and the second element of the block containing

$1$ .

Matching statistic: St000010

Mapping

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 88% ●values known / values provided: 100%●distinct values known / distinct values provided: 88%

Values

[1,0]
 => [1,0]
 => [1] => [1]
 => 1
[1,0,1,0]
 => [1,1,0,0]
 => [2] => [2]
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1] => [1,1]
 => 2
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3] => [3]
 => 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [3] => [3]
 => 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2,1] => [2,1]
 => 2
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,2] => [2,1]
 => 2
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1] => [1,1,1]
 => 3
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => [4]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4] => [4]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [4] => [4]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [4] => [4]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [4] => [4]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => [3,1]
 => 2
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1] => [3,1]
 => 2
[1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => [2,2]
 => 2
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => [3,1]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => [3,1]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => [2,1,1]
 => 3
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,1,1]
 => 3
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1,1]
 => 3
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => [5]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5] => [5]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5] => [5]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5] => [5]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [5] => [5]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [5] => [5]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [5] => [5]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [5] => [5]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [5] => [5]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [5] => [5]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [5] => [5]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [5] => [5]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [5] => [5]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5] => [5]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => [4,1]
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1] => [4,1]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1] => [4,1]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [4,1]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [4,1]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [3,2]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [3,2]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [3,2]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [4,1]
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [4,1]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [3,2]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [4,1]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [4,1]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [4,1]
 => 2
[]
 => []
 => [] => ?
 => ? = 0

Description

The length of the partition.

The following 156 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000068The number of minimal elements in a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000505The biggest entry in the block containing the 1. St001050The number of terminal closers of a set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001494The Alon-Tarsi number of a graph. St001581The achromatic number of a graph. St001733The number of weak left to right maxima of a Dyck path. St000053The number of valleys of the Dyck path. St000234The number of global ascents of a permutation. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St000925The number of topologically connected components of a set partition. St000502The number of successions of a set partitions. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St000363The number of minimal vertex covers of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001316The domatic number of a graph. St000069The number of maximal elements of a poset. St000738The first entry in the last row of a standard tableau. St000617The number of global maxima of a Dyck path. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000654The first descent of a permutation. St000906The length of the shortest maximal chain in a poset. St000740The last entry of a permutation. St000717The number of ordinal summands of a poset. St000989The number of final rises of a permutation. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St001808The box weight or horizontal decoration of a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000947The major index east count of a Dyck path. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000054The first entry of the permutation. St000007The number of saliances of the permutation. St000546The number of global descents of a permutation. St000702The number of weak deficiencies of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000542The number of left-to-right-minima 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. St000759The smallest missing part in an integer partition. St000203The number of external nodes of a binary tree. St000237The number of small exceedances. St000990The first ascent of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000993The multiplicity of the largest part of an integer partition. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000843The decomposition number of a perfect matching. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000734The last entry in the first row of a standard tableau. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000314The number of left-to-right-maxima of a permutation. St000352The Elizalde-Pak rank of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000051The size of the left subtree of a binary tree. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. 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. St000015The number of peaks of a Dyck path. St000213The number of weak exceedances (also weak excedences) of a permutation. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000335The difference of lower and upper interactions. St000374The number of exclusive right-to-left minima of a permutation. St000822The Hadwiger number of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St001202Call 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001481The minimal height of a peak of a Dyck path. St000005The bounce statistic of a Dyck path. St000120The number of left tunnels of a Dyck path. St000133The "bounce" of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000331The number of upper interactions of a Dyck path. St000883The number of longest increasing subsequences of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000061The number of nodes on the left branch of a binary tree. St001812The biclique partition number of a graph. St001330The hat guessing number of a graph. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000160The multiplicity of the smallest part of a partition. St000475The number of parts equal to 1 in a partition. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000924The number of topologically connected components of a perfect matching. St001497The position of the largest weak excedence of a permutation. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000181The number of connected components of the Hasse diagram for the poset. St000338The number of pixed points of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000145The Dyson rank of a partition. St001889The size of the connectivity set of a signed permutation. St001566The length of the longest arithmetic progression in a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001434The number of negative sum pairs of a signed permutation. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St000392The length of the longest run of ones in a binary word. St001372The length of a longest cyclic run of ones of a binary word. St000444The length of the maximal rise of a Dyck path. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001937The size of the center of a parking function. St000942The number of critical left to right maxima of the parking functions. St001904The length of the initial strictly increasing segment of a parking function. St001557The number of inversions of the second entry of a permutation. St001621The number of atoms of a lattice.