- FindStat

Your data matches 218 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000228
(load all 331 compositions to match this statistic)

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1,2,3}}
 => [3]
 => 3
{{1},{2},{3}}
 => [1,1,1]
 => 3
{{1,2,3,4}}
 => [4]
 => 4
{{1,2},{3,4}}
 => [2,2]
 => 4
{{1,3},{2,4}}
 => [2,2]
 => 4
{{1,4},{2,3}}
 => [2,2]
 => 4
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => 4
{{1,2,3,4,5}}
 => [5]
 => 5
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => 5
{{1,2,3,4,5,6}}
 => [6]
 => 6
{{1,2,3},{4,5,6}}
 => [3,3]
 => 6
{{1,2,4},{3,5,6}}
 => [3,3]
 => 6
{{1,2,5},{3,4,6}}
 => [3,3]
 => 6
{{1,2,6},{3,4,5}}
 => [3,3]
 => 6
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => 6
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => 6
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => 6
{{1,3,4},{2,5,6}}
 => [3,3]
 => 6
{{1,3,5},{2,4,6}}
 => [3,3]
 => 6
{{1,3,6},{2,4,5}}
 => [3,3]
 => 6
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => 6
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => 6
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => 6
{{1,4,5},{2,3,6}}
 => [3,3]
 => 6
{{1,4,6},{2,3,5}}
 => [3,3]
 => 6
{{1,4},{2,3},{5,6}}
 => [2,2,2]
 => 6
{{1,5,6},{2,3,4}}
 => [3,3]
 => 6
{{1,5},{2,3},{4,6}}
 => [2,2,2]
 => 6
{{1,6},{2,3},{4,5}}
 => [2,2,2]
 => 6
{{1,4},{2,5},{3,6}}
 => [2,2,2]
 => 6
{{1,4},{2,6},{3,5}}
 => [2,2,2]
 => 6
{{1,5},{2,4},{3,6}}
 => [2,2,2]
 => 6
{{1,6},{2,4},{3,5}}
 => [2,2,2]
 => 6
{{1,5},{2,6},{3,4}}
 => [2,2,2]
 => 6
{{1,6},{2,5},{3,4}}
 => [2,2,2]
 => 6
{{1},{2},{3},{4},{5},{6}}
 => [1,1,1,1,1,1]
 => 6
{{1,2,3,4,5,6,7}}
 => [7]
 => 7
{{1},{2},{3},{4},{5},{6},{7}}
 => [1,1,1,1,1,1,1]
 => 7

Description

The size of a partition. This statistic is the constant statistic of the level sets.

Matching statistic: St000293
(load all 19 compositions to match this statistic)

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1,2,3}}
 => [3]
 => 1000 => 3
{{1},{2},{3}}
 => [1,1,1]
 => 1110 => 3
{{1,2,3,4}}
 => [4]
 => 10000 => 4
{{1,2},{3,4}}
 => [2,2]
 => 1100 => 4
{{1,3},{2,4}}
 => [2,2]
 => 1100 => 4
{{1,4},{2,3}}
 => [2,2]
 => 1100 => 4
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => 11110 => 4
{{1,2,3,4,5}}
 => [5]
 => 100000 => 5
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => 111110 => 5
{{1,2,3,4,5,6}}
 => [6]
 => 1000000 => 6
{{1,2,3},{4,5,6}}
 => [3,3]
 => 11000 => 6
{{1,2,4},{3,5,6}}
 => [3,3]
 => 11000 => 6
{{1,2,5},{3,4,6}}
 => [3,3]
 => 11000 => 6
{{1,2,6},{3,4,5}}
 => [3,3]
 => 11000 => 6
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => 11100 => 6
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => 11100 => 6
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => 11100 => 6
{{1,3,4},{2,5,6}}
 => [3,3]
 => 11000 => 6
{{1,3,5},{2,4,6}}
 => [3,3]
 => 11000 => 6
{{1,3,6},{2,4,5}}
 => [3,3]
 => 11000 => 6
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => 11100 => 6
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => 11100 => 6
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => 11100 => 6
{{1,4,5},{2,3,6}}
 => [3,3]
 => 11000 => 6
{{1,4,6},{2,3,5}}
 => [3,3]
 => 11000 => 6
{{1,4},{2,3},{5,6}}
 => [2,2,2]
 => 11100 => 6
{{1,5,6},{2,3,4}}
 => [3,3]
 => 11000 => 6
{{1,5},{2,3},{4,6}}
 => [2,2,2]
 => 11100 => 6
{{1,6},{2,3},{4,5}}
 => [2,2,2]
 => 11100 => 6
{{1,4},{2,5},{3,6}}
 => [2,2,2]
 => 11100 => 6
{{1,4},{2,6},{3,5}}
 => [2,2,2]
 => 11100 => 6
{{1,5},{2,4},{3,6}}
 => [2,2,2]
 => 11100 => 6
{{1,6},{2,4},{3,5}}
 => [2,2,2]
 => 11100 => 6
{{1,5},{2,6},{3,4}}
 => [2,2,2]
 => 11100 => 6
{{1,6},{2,5},{3,4}}
 => [2,2,2]
 => 11100 => 6
{{1},{2},{3},{4},{5},{6}}
 => [1,1,1,1,1,1]
 => 1111110 => 6
{{1,2,3,4,5,6,7}}
 => [7]
 => 10000000 => 7
{{1},{2},{3},{4},{5},{6},{7}}
 => [1,1,1,1,1,1,1]
 => 11111110 => 7

Description

The number of inversions of a binary word.

Matching statistic: St000395
(load all 17 compositions to match this statistic)

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => 3
{{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => 3
{{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => 4
{{1,2},{3,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 4
{{1,3},{2,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 4
{{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 4
{{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
{{1,2,3,4,5,6}}
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
{{1,2,3},{4,5,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,2,4},{3,5,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,2,5},{3,4,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,2,6},{3,4,5}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,2},{3,4},{5,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,2},{3,5},{4,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,2},{3,6},{4,5}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,3,4},{2,5,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,3,5},{2,4,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,3,6},{2,4,5}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,3},{2,4},{5,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,3},{2,5},{4,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,3},{2,6},{4,5}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,4,5},{2,3,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,4,6},{2,3,5}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,4},{2,3},{5,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,5,6},{2,3,4}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,5},{2,3},{4,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,6},{2,3},{4,5}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,4},{2,5},{3,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,4},{2,6},{3,5}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,5},{2,4},{3,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,6},{2,4},{3,5}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,5},{2,6},{3,4}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,6},{2,5},{3,4}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1},{2},{3},{4},{5},{6}}
 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
{{1,2,3,4,5,6,7}}
 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 7
{{1},{2},{3},{4},{5},{6},{7}}
 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7

Description

The sum of the heights of the peaks of a Dyck path.

Matching statistic: St000645
(load all 3 compositions to match this statistic)

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1,2,3}}
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3
{{1},{2},{3}}
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
{{1,2,3,4}}
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
{{1,2},{3,4}}
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 4
{{1,3},{2,4}}
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 4
{{1,4},{2,3}}
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 4
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
{{1,2,3,4,5}}
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
{{1,2,3,4,5,6}}
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 6
{{1,2,3},{4,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
{{1,2,4},{3,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
{{1,2,5},{3,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
{{1,2,6},{3,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
{{1,3,4},{2,5,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
{{1,3,5},{2,4,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
{{1,3,6},{2,4,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
{{1,4,5},{2,3,6}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
{{1,4,6},{2,3,5}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
{{1,4},{2,3},{5,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
{{1,5,6},{2,3,4}}
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
{{1,5},{2,3},{4,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
{{1,6},{2,3},{4,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
{{1,4},{2,5},{3,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
{{1,4},{2,6},{3,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
{{1,5},{2,4},{3,6}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
{{1,6},{2,4},{3,5}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
{{1,5},{2,6},{3,4}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
{{1,6},{2,5},{3,4}}
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
{{1},{2},{3},{4},{5},{6}}
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
{{1,2,3,4,5,6,7}}
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => 7
{{1},{2},{3},{4},{5},{6},{7}}
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 7

Description

The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. For a Dyck path

$D = D_1 \cdots D_{2n}$ with peaks in positions

$i_1 < \ldots < i_k$ and valleys in positions

$j_1 < \ldots < j_{k-1}$ , this statistic is given by

$\sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a)$

Matching statistic: St001020
(load all 40 compositions to match this statistic)

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => 3
{{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => 3
{{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => 4
{{1,2},{3,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 4
{{1,3},{2,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 4
{{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 4
{{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
{{1,2,3,4,5,6}}
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
{{1,2,3},{4,5,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,2,4},{3,5,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,2,5},{3,4,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,2,6},{3,4,5}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,2},{3,4},{5,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,2},{3,5},{4,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,2},{3,6},{4,5}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,3,4},{2,5,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,3,5},{2,4,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,3,6},{2,4,5}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,3},{2,4},{5,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,3},{2,5},{4,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,3},{2,6},{4,5}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,4,5},{2,3,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,4,6},{2,3,5}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,4},{2,3},{5,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,5,6},{2,3,4}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,5},{2,3},{4,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,6},{2,3},{4,5}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,4},{2,5},{3,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,4},{2,6},{3,5}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,5},{2,4},{3,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,6},{2,4},{3,5}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,5},{2,6},{3,4}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,6},{2,5},{3,4}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1},{2},{3},{4},{5},{6}}
 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
{{1,2,3,4,5,6,7}}
 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 7
{{1},{2},{3},{4},{5},{6},{7}}
 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7

Description

Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001034
(load all 20 compositions to match this statistic)

Mapping

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1,2,3}}
 => [3]
 => [1,0,1,0,1,0]
 => 3
{{1},{2},{3}}
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 3
{{1,2,3,4}}
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
{{1,2},{3,4}}
 => [2,2]
 => [1,1,1,0,0,0]
 => 4
{{1,3},{2,4}}
 => [2,2]
 => [1,1,1,0,0,0]
 => 4
{{1,4},{2,3}}
 => [2,2]
 => [1,1,1,0,0,0]
 => 4
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 4
{{1,2,3,4,5}}
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5
{{1,2,3,4,5,6}}
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
{{1,2,3},{4,5,6}}
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 6
{{1,2,4},{3,5,6}}
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 6
{{1,2,5},{3,4,6}}
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 6
{{1,2,6},{3,4,5}}
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 6
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 6
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 6
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 6
{{1,3,4},{2,5,6}}
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 6
{{1,3,5},{2,4,6}}
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 6
{{1,3,6},{2,4,5}}
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 6
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 6
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 6
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 6
{{1,4,5},{2,3,6}}
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 6
{{1,4,6},{2,3,5}}
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 6
{{1,4},{2,3},{5,6}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 6
{{1,5,6},{2,3,4}}
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 6
{{1,5},{2,3},{4,6}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 6
{{1,6},{2,3},{4,5}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 6
{{1,4},{2,5},{3,6}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 6
{{1,4},{2,6},{3,5}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 6
{{1,5},{2,4},{3,6}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 6
{{1,6},{2,4},{3,5}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 6
{{1,5},{2,6},{3,4}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 6
{{1,6},{2,5},{3,4}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 6
{{1},{2},{3},{4},{5},{6}}
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 6
{{1,2,3,4,5,6,7}}
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7
{{1},{2},{3},{4},{5},{6},{7}}
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 7

Description

The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].

Matching statistic: St001437
(load all 3 compositions to match this statistic)

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St001437: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1,2,3}}
 => [3] => 100 => 3
{{1},{2},{3}}
 => [1,1,1] => 111 => 3
{{1,2,3,4}}
 => [4] => 1000 => 4
{{1,2},{3,4}}
 => [2,2] => 1010 => 4
{{1,3},{2,4}}
 => [2,2] => 1010 => 4
{{1,4},{2,3}}
 => [2,2] => 1010 => 4
{{1},{2},{3},{4}}
 => [1,1,1,1] => 1111 => 4
{{1,2,3,4,5}}
 => [5] => 10000 => 5
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1] => 11111 => 5
{{1,2,3,4,5,6}}
 => [6] => 100000 => 6
{{1,2,3},{4,5,6}}
 => [3,3] => 100100 => 6
{{1,2,4},{3,5,6}}
 => [3,3] => 100100 => 6
{{1,2,5},{3,4,6}}
 => [3,3] => 100100 => 6
{{1,2,6},{3,4,5}}
 => [3,3] => 100100 => 6
{{1,2},{3,4},{5,6}}
 => [2,2,2] => 101010 => 6
{{1,2},{3,5},{4,6}}
 => [2,2,2] => 101010 => 6
{{1,2},{3,6},{4,5}}
 => [2,2,2] => 101010 => 6
{{1,3,4},{2,5,6}}
 => [3,3] => 100100 => 6
{{1,3,5},{2,4,6}}
 => [3,3] => 100100 => 6
{{1,3,6},{2,4,5}}
 => [3,3] => 100100 => 6
{{1,3},{2,4},{5,6}}
 => [2,2,2] => 101010 => 6
{{1,3},{2,5},{4,6}}
 => [2,2,2] => 101010 => 6
{{1,3},{2,6},{4,5}}
 => [2,2,2] => 101010 => 6
{{1,4,5},{2,3,6}}
 => [3,3] => 100100 => 6
{{1,4,6},{2,3,5}}
 => [3,3] => 100100 => 6
{{1,4},{2,3},{5,6}}
 => [2,2,2] => 101010 => 6
{{1,5,6},{2,3,4}}
 => [3,3] => 100100 => 6
{{1,5},{2,3},{4,6}}
 => [2,2,2] => 101010 => 6
{{1,6},{2,3},{4,5}}
 => [2,2,2] => 101010 => 6
{{1,4},{2,5},{3,6}}
 => [2,2,2] => 101010 => 6
{{1,4},{2,6},{3,5}}
 => [2,2,2] => 101010 => 6
{{1,5},{2,4},{3,6}}
 => [2,2,2] => 101010 => 6
{{1,6},{2,4},{3,5}}
 => [2,2,2] => 101010 => 6
{{1,5},{2,6},{3,4}}
 => [2,2,2] => 101010 => 6
{{1,6},{2,5},{3,4}}
 => [2,2,2] => 101010 => 6
{{1},{2},{3},{4},{5},{6}}
 => [1,1,1,1,1,1] => 111111 => 6
{{1,2,3,4,5,6,7}}
 => [7] => 1000000 => 7
{{1},{2},{3},{4},{5},{6},{7}}
 => [1,1,1,1,1,1,1] => 1111111 => 7

Description

The flex of a binary word. This is the product of the lex statistic ([[St001436]], augmented by 1) and its frequency ([[St000627]]), see [1, §8].

Matching statistic: St001523
(load all 9 compositions to match this statistic)

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001523: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => 3
{{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => 3
{{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => 4
{{1,2},{3,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 4
{{1,3},{2,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 4
{{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 4
{{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
{{1,2,3,4,5,6}}
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
{{1,2,3},{4,5,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,2,4},{3,5,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,2,5},{3,4,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,2,6},{3,4,5}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,2},{3,4},{5,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,2},{3,5},{4,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,2},{3,6},{4,5}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,3,4},{2,5,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,3,5},{2,4,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,3,6},{2,4,5}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,3},{2,4},{5,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,3},{2,5},{4,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,3},{2,6},{4,5}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,4,5},{2,3,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,4,6},{2,3,5}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,4},{2,3},{5,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,5,6},{2,3,4}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 6
{{1,5},{2,3},{4,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,6},{2,3},{4,5}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,4},{2,5},{3,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,4},{2,6},{3,5}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,5},{2,4},{3,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,6},{2,4},{3,5}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,5},{2,6},{3,4}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1,6},{2,5},{3,4}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 6
{{1},{2},{3},{4},{5},{6}}
 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
{{1,2,3,4,5,6,7}}
 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 7
{{1},{2},{3},{4},{5},{6},{7}}
 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7

Description

The degree of symmetry of a Dyck path. Given a Dyck path

$D=(d_1,\dots,d_{2n})$ , with

$d_i\in\{0,1\}$ , this is the number of positions

$1\leq i\leq n$ such that

$d_i = 1-d_{2n+1-i}$ and the initial height of the

$i$ -th step

$\sum_{j < i} d_i$ equals the final height of the

$(2n+1-i)$ -th step.

Matching statistic: St000998
(load all 15 compositions to match this statistic)

Mapping

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => 4 = 3 + 1
{{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => 4 = 3 + 1
{{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
{{1,2},{3,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
{{1,3},{2,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
{{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
{{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
{{1,2,3,4,5,6}}
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 7 = 6 + 1
{{1,2,3},{4,5,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 7 = 6 + 1
{{1,2,4},{3,5,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 7 = 6 + 1
{{1,2,5},{3,4,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 7 = 6 + 1
{{1,2,6},{3,4,5}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 7 = 6 + 1
{{1,2},{3,4},{5,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 7 = 6 + 1
{{1,2},{3,5},{4,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 7 = 6 + 1
{{1,2},{3,6},{4,5}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 7 = 6 + 1
{{1,3,4},{2,5,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 7 = 6 + 1
{{1,3,5},{2,4,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 7 = 6 + 1
{{1,3,6},{2,4,5}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 7 = 6 + 1
{{1,3},{2,4},{5,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 7 = 6 + 1
{{1,3},{2,5},{4,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 7 = 6 + 1
{{1,3},{2,6},{4,5}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 7 = 6 + 1
{{1,4,5},{2,3,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 7 = 6 + 1
{{1,4,6},{2,3,5}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 7 = 6 + 1
{{1,4},{2,3},{5,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 7 = 6 + 1
{{1,5,6},{2,3,4}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 7 = 6 + 1
{{1,5},{2,3},{4,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 7 = 6 + 1
{{1,6},{2,3},{4,5}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 7 = 6 + 1
{{1,4},{2,5},{3,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 7 = 6 + 1
{{1,4},{2,6},{3,5}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 7 = 6 + 1
{{1,5},{2,4},{3,6}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 7 = 6 + 1
{{1,6},{2,4},{3,5}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 7 = 6 + 1
{{1,5},{2,6},{3,4}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 7 = 6 + 1
{{1,6},{2,5},{3,4}}
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 7 = 6 + 1
{{1},{2},{3},{4},{5},{6}}
 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
{{1,2,3,4,5,6,7}}
 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 8 = 7 + 1
{{1},{2},{3},{4},{5},{6},{7}}
 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 8 = 7 + 1

Description

Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001332
(load all 58 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001332: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1,2,3}}
 => [2,3,1] => [1,2,3] => 2 = 3 - 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 2 = 3 - 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => 3 = 4 - 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,2,3,4] => 3 = 4 - 1
{{1,3},{2,4}}
 => [3,4,1,2] => [1,3,2,4] => 3 = 4 - 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,4,2,3] => 3 = 4 - 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => 4 = 5 - 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 5 = 6 - 1
{{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => [1,2,3,4,5,6] => 5 = 6 - 1
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => [1,2,4,3,5,6] => 5 = 6 - 1
{{1,2,5},{3,4,6}}
 => [2,5,4,6,1,3] => [1,2,5,3,4,6] => 5 = 6 - 1
{{1,2,6},{3,4,5}}
 => [2,6,4,5,3,1] => [1,2,6,3,4,5] => 5 = 6 - 1
{{1,2},{3,4},{5,6}}
 => [2,1,4,3,6,5] => [1,2,3,4,5,6] => 5 = 6 - 1
{{1,2},{3,5},{4,6}}
 => [2,1,5,6,3,4] => [1,2,3,5,4,6] => 5 = 6 - 1
{{1,2},{3,6},{4,5}}
 => [2,1,6,5,4,3] => [1,2,3,6,4,5] => 5 = 6 - 1
{{1,3,4},{2,5,6}}
 => [3,5,4,1,6,2] => [1,3,4,2,5,6] => 5 = 6 - 1
{{1,3,5},{2,4,6}}
 => [3,4,5,6,1,2] => [1,3,5,2,4,6] => 5 = 6 - 1
{{1,3,6},{2,4,5}}
 => [3,4,6,5,2,1] => [1,3,6,2,4,5] => 5 = 6 - 1
{{1,3},{2,4},{5,6}}
 => [3,4,1,2,6,5] => [1,3,2,4,5,6] => 5 = 6 - 1
{{1,3},{2,5},{4,6}}
 => [3,5,1,6,2,4] => [1,3,2,5,4,6] => 5 = 6 - 1
{{1,3},{2,6},{4,5}}
 => [3,6,1,5,4,2] => [1,3,2,6,4,5] => 5 = 6 - 1
{{1,4,5},{2,3,6}}
 => [4,3,6,5,1,2] => [1,4,5,2,3,6] => 5 = 6 - 1
{{1,4,6},{2,3,5}}
 => [4,3,5,6,2,1] => [1,4,6,2,3,5] => 5 = 6 - 1
{{1,4},{2,3},{5,6}}
 => [4,3,2,1,6,5] => [1,4,2,3,5,6] => 5 = 6 - 1
{{1,5,6},{2,3,4}}
 => [5,3,4,2,6,1] => [1,5,6,2,3,4] => 5 = 6 - 1
{{1,5},{2,3},{4,6}}
 => [5,3,2,6,1,4] => [1,5,2,3,4,6] => 5 = 6 - 1
{{1,6},{2,3},{4,5}}
 => [6,3,2,5,4,1] => [1,6,2,3,4,5] => 5 = 6 - 1
{{1,4},{2,5},{3,6}}
 => [4,5,6,1,2,3] => [1,4,2,5,3,6] => 5 = 6 - 1
{{1,4},{2,6},{3,5}}
 => [4,6,5,1,3,2] => [1,4,2,6,3,5] => 5 = 6 - 1
{{1,5},{2,4},{3,6}}
 => [5,4,6,2,1,3] => [1,5,2,4,3,6] => 5 = 6 - 1
{{1,6},{2,4},{3,5}}
 => [6,4,5,2,3,1] => [1,6,2,4,3,5] => 5 = 6 - 1
{{1,5},{2,6},{3,4}}
 => [5,6,4,3,1,2] => [1,5,2,6,3,4] => 5 = 6 - 1
{{1,6},{2,5},{3,4}}
 => [6,5,4,3,2,1] => [1,6,2,5,3,4] => 5 = 6 - 1
{{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 5 = 6 - 1
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 6 = 7 - 1
{{1},{2},{3},{4},{5},{6},{7}}
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 6 = 7 - 1

Description

The number of steps on the non-negative side of the walk associated with the permutation. Consider the walk taking an up step for each ascent, and a down step for each descent of the permutation. Then this statistic is the number of steps that begin and end at non-negative height.

The following 208 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St000018The number of inversions of a permutation. St000054The first entry of the permutation. St000246The number of non-inversions of a permutation. St000290The major index of a binary word. St000294The number of distinct factors of a binary word. St000296The length of the symmetric border of a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000518The number of distinct subsequences in a binary word. St000543The size of the conjugacy class of a binary word. St000548The number of different non-empty partial sums of an integer partition. St000564The number of occurrences of the pattern {{1},{2}} in a set partition. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000734The last entry in the first row of a standard tableau. St000806The semiperimeter of the associated bargraph. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000883The number of longest increasing subsequences of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001746The coalition number of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000019The cardinality of the support of a permutation. St000081The number of edges of a graph. St000141The maximum drop size of a permutation. St000189The number of elements in the poset. St000393The number of strictly increasing runs in a binary word. St000553The number of blocks of a graph. St000627The exponent of a binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001371The length of the longest Yamanouchi prefix of a binary word. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001430The number of positive entries in a signed permutation. St001479The number of bridges of a graph. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001884The number of borders of a binary word. St001917The order of toric promotion on the set of labellings of a graph. St000295The length of the border of a binary word. St000519The largest length of a factor maximising the subword complexity. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000110The number of permutations less than or equal to a permutation in left weak order. St000890The number of nonzero entries in an alternating sign matrix. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001958The degree of the polynomial interpolating the values of a permutation. St001925The minimal number of zeros in a row of an alternating sign matrix. St000144The pyramid weight of the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001622The number of join-irreducible elements of a lattice. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000064The number of one-box pattern of a permutation. St000171The degree of the graph. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000235The number of indices that are not cyclical small weak excedances. St000240The number of indices that are not small excedances. St000288The number of ones in a binary word. St000336The leg major index of a standard tableau. St000501The size of the first part in the decomposition of a permutation. St000656The number of cuts of a poset. St000844The size of the largest block in the direct sum decomposition of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001717The largest size of an interval in a poset. St000026The position of the first return of a Dyck path. St000058The order of a permutation. St000060The greater neighbor of the maximum. St000167The number of leaves of an ordered tree. St000197The number of entries equal to positive one in the alternating sign matrix. St000209Maximum difference of elements in cycles. St000316The number of non-left-to-right-maxima of a permutation. St000653The last descent of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000956The maximal displacement of a permutation. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. 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. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St000829The Ulam distance of a permutation to the identity permutation. St001759The Rajchgot index of a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St001911A descent variant minus the number of inversions. St001959The product of the heights of the peaks of a Dyck path. St000029The depth of a permutation. St000809The reduced reflection length of the permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000719The number of alignments in a perfect matching. St001684The reduced word complexity of a permutation. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St001725The harmonious chromatic number of a graph. St001645The pebbling number of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000924The number of topologically connected components of a perfect matching. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000672The number of minimal elements in Bruhat order not less than the permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000528The height of a poset. St000906The length of the shortest maximal chain in a poset. St000080The rank of the poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St001077The prefix exchange distance of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001429The number of negative entries in a signed permutation. St001480The number of simple summands of the module J^2/J^3. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000327The number of cover relations in a poset. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000673The number of non-fixed points of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001637The number of (upper) dissectors of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St001927Sparre Andersen's number of positives of a signed permutation. St000837The number of ascents of distance 2 of a permutation. St001468The smallest fixpoint of a permutation. St001557The number of inversions of the second entry of a permutation. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001817The number of flag weak exceedances of a signed permutation. St001892The flag excedance statistic of a signed permutation. St001434The number of negative sum pairs of a signed permutation. St000135The number of lucky cars of the parking function. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000744The length of the path to the largest entry in a standard Young tableau. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St000044The number of vertices of the unicellular map given by a perfect matching. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001926Sparre Andersen's position of the maximum of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001668The number of points of the poset minus the width of the poset. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001488The number of corners of a skew partition. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000075The orbit size of a standard tableau under promotion. St000454The largest eigenvalue of a graph if it is integral. St001424The number of distinct squares in a binary word. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000391The sum of the positions of the ones in a binary word. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001621The number of atoms of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000522The number of 1-protected nodes of a rooted tree. St000521The number of distinct subtrees of an ordered tree. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. 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

$eA$ . St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.