- FindStat

Your data matches 245 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000439
(load all 25 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St000025
(load all 25 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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: St000839
(load all 77 compositions to match this statistic)

Mapping

Mp00151: Permutations —to cycle type⟶ Set partitions
St000839: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest opener of a set partition. An opener (or left hand endpoint) of a set partition is a number that is minimal in its block. For this statistic, singletons are considered as openers.

Matching statistic: St000971
(load all 47 compositions to match this statistic)

Mapping

Mp00151: Permutations —to cycle type⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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: St000011
(load all 22 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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: St000026
(load all 6 compositions to match this statistic)

Mapping

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [.,.]
 => [1,0]
 => 1 = 2 - 1
[1,2] => [.,[.,.]]
 => [1,0,1,0]
 => 1 = 2 - 1
[2,1] => [[.,.],.]
 => [1,1,0,0]
 => 2 = 3 - 1
[1,2,3] => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 1 = 2 - 1
[1,3,2] => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[2,1,3] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 2 = 3 - 1
[2,3,1] => [[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => 3 = 4 - 1
[3,1,2] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 2 = 3 - 1
[3,2,1] => [[[.,.],.],.]
 => [1,1,1,0,0,0]
 => 3 = 4 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 3 = 4 - 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => 4 = 5 - 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 3 = 4 - 1
[2,4,3,1] => [[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => 4 = 5 - 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[3,1,4,2] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[3,2,4,1] => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => 4 = 5 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 3 = 4 - 1
[3,4,2,1] => [[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => 4 = 5 - 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[4,1,3,2] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[4,2,1,3] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => 4 = 5 - 1
[4,3,1,2] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => 4 = 5 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1

Description

The position of the first return of a Dyck path.

Matching statistic: St000382
(load all 9 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to 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] => [1,0]
 => [1] => 1 = 2 - 1
[1,2] => [1,0,1,0]
 => [1,1] => 1 = 2 - 1
[2,1] => [1,1,0,0]
 => [2] => 2 = 3 - 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,1] => 1 = 2 - 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,2] => 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0]
 => [2,1] => 2 = 3 - 1
[2,3,1] => [1,1,0,1,0,0]
 => [2,1] => 2 = 3 - 1
[3,1,2] => [1,1,1,0,0,0]
 => [3] => 3 = 4 - 1
[3,2,1] => [1,1,1,0,0,0]
 => [3] => 3 = 4 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1 = 2 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,2] => 1 = 2 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,2,1] => 1 = 2 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,2,1] => 1 = 2 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,3] => 1 = 2 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,3] => 1 = 2 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [2,1,1] => 2 = 3 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2] => 2 = 3 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [2,1,1] => 2 = 3 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,1,1] => 2 = 3 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [2,2] => 2 = 3 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [2,2] => 2 = 3 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3,1] => 3 = 4 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [3,1] => 3 = 4 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3,1] => 3 = 4 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [3,1] => 3 = 4 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [3,1] => 3 = 4 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [3,1] => 3 = 4 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [4] => 4 = 5 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [4] => 4 = 5 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [4] => 4 = 5 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [4] => 4 = 5 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [4] => 4 = 5 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [4] => 4 = 5 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 1 = 2 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 1 = 2 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 1 = 2 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => 1 = 2 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 1 = 2 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 1 = 2 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 1 = 2 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => 1 = 2 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => 1 = 2 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => 1 = 2 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => 1 = 2 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 1 = 2 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => 1 = 2 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 1 = 2 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => 1 = 2 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => 1 = 2 - 1

Description

The first part of an integer composition.

Matching statistic: St000505
(load all 3 compositions to match this statistic)

Mapping

Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00091: Set partitions —rotate increasing⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The biggest entry in the block containing the 1.

Matching statistic: St000738
(load all 12 compositions to match this statistic)

Mapping

Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].

Matching statistic: St001784
(load all 8 compositions to match this statistic)

Mapping

Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00249: Set partitions —Callan switch⟶ Set partitions
St001784: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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

$1$ .

The following 235 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001161The major index north count of a Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000013The height of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000383The last part of an integer composition. St000676The number of odd rises of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. 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. St001733The number of weak left to right maxima of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000008The major index of the composition. St000010The length of the partition. St000012The area of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000160The multiplicity of the smallest part of a partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000548The number of different non-empty partial sums of an integer partition. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000504The cardinality of the first block of a set partition. St000693The modular (standard) major index of a standard tableau. St000678The number of up steps after the last double rise of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000502The number of successions of a set partitions. St000326The position of the first one in a binary word after appending a 1 at the end. St000444The length of the maximal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000297The number of leading ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000730The maximal arc length of a set partition. St000874The position of the last double rise in a Dyck path. St000947The major index east count of a Dyck path. St000984The number of boxes below precisely one peak. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000996The number of exclusive left-to-right maxima of a permutation. St000147The largest part of an integer partition. St000234The number of global ascents of a permutation. St001759The Rajchgot index of a permutation. St000069The number of maximal elements of a poset. St000378The diagonal inversion number of an integer partition. St000068The number of minimal elements in a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000028The number of stack-sorts needed to sort a permutation. St000740The last entry of a permutation. St000446The disorder of a permutation. St000702The number of weak deficiencies of a permutation. St000798The makl of a permutation. St000617The number of global maxima of a Dyck path. St000759The smallest missing part in an integer partition. St001461The number of topologically connected components of the chord diagram of a permutation. St000654The first descent of a permutation. St000957The number of Bruhat lower covers of a permutation. St000653The last descent of a permutation. St000989The number of final rises of a permutation. St000054The first entry of the permutation. St000141The maximum drop size of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000007The number of saliances of the permutation. St000717The number of ordinal summands of a poset. St000019The cardinality of the support of a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000956The maximal displacement of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000237The number of small exceedances. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000546The number of global descents of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001671Haglund's hag of a permutation. St000797The stat`` of a permutation. St001497The position of the largest weak excedence of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000259The diameter of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000203The number of external nodes of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000161The sum of the sizes of the right subtrees of a binary tree. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000794The mak of a permutation. St000809The reduced reflection length of the permutation. St000833The comajor index of a permutation. St000501The size of the first part in the decomposition of a permutation. St001645The pebbling number of a connected graph. St000843The decomposition number of a perfect matching. St000067The inversion number of the alternating sign matrix. St000260The radius of a connected graph. 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. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St000990The first ascent of a permutation. St001875The number of simple modules with projective dimension at most 1. St001806The upper middle entry of a permutation. St001807The lower middle entry of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000051The size of the left subtree of a binary tree. St000133The "bounce" of a permutation. 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. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension 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. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000240The number of indices that are not small excedances. St000352The Elizalde-Pak rank of a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St000004The major index of a permutation. St000005The bounce statistic of a Dyck path. St000120The number of left tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000204The number of internal nodes of a binary tree. St000304The load of a permutation. St000305The inverse major index of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000339The maf index of a permutation. St000356The number of occurrences of the pattern 13-2. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. 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. St001480The number of simple summands of the module J^2/J^3. 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. St000082The number of elements smaller than a binary tree in Tamari order. St000692Babson and Steingrímsson's statistic of a permutation. St000216The absolute length of a permutation. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000532The total number of rook placements on a Ferrers board. St001400The total number of Littlewood-Richardson tableaux of given shape. St001814The number of partitions interlacing the given partition. St000184The size of the centralizer of any permutation of given cycle type. St000228The size of a partition. St000384The maximal part of the shifted composition of an integer partition. St000391The sum of the positions of the ones in a binary word. 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. St000531The leading coefficient of the rook polynomial of an integer partition. St000667The greatest common divisor of the parts of the partition. St000784The maximum of the length and the largest part of the integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001267The length of the Lyndon factorization of the binary word. St001313The number of Dyck paths above the lattice path given by a binary word. St001360The number of covering relations in Young's lattice below a partition. St001389The number of partitions of the same length below the given integer partition. St001437The flex of a binary word. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St000145The Dyson rank of a partition. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000682The Grundy value of Welter's game on a binary word. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000840The number of closers smaller than the largest opener in a perfect matching. 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. St001615The number of join prime elements of a lattice. 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. St001434The number of negative sum pairs of a signed permutation. St000060The greater neighbor of the maximum. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St000461The rix statistic of a permutation. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001430The number of positive entries in a signed permutation. St000455The second largest eigenvalue of a graph if it is integral. St001115The number of even descents of a permutation. St001394The genus of a permutation. St001429The number of negative entries in a signed permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. 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

$eA$ . St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000906The length of the shortest maximal chain in a poset. St000115The single entry in the last row. St000942The number of critical left to right maxima of the parking functions. 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. 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. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001557The number of inversions of the second entry of a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001927Sparre Andersen's number of positives of a signed permutation. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001877Number of indecomposable injective modules with projective dimension 2.