- FindStat

Your data matches 139 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000011
(load all 48 compositions to match this statistic)

Mapping

Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => 1
[1,0,1,0]
 => [1,0,1,0]
 => 2
[1,1,0,0]
 => [1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 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: St000678
(load all 48 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St001050
(load all 28 compositions to match this statistic)

Mapping

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of terminal closers of a set partition. A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.

Matching statistic: St001184
(load all 80 compositions to match this statistic)

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.

Matching statistic: St000007
(load all 61 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern

$([1], {(1,1)})$ , i.e., the upper right quadrant is shaded, see [1].

Matching statistic: St000025
(load all 30 compositions to match this statistic)

Mapping

Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => [1,0]
 => 1
[1,0,1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 2
[1,1,0,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 3
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 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: St000056
(load all 46 compositions to match this statistic)

Mapping

Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The decomposition (or block) number of a permutation. For

$\pi \in \mathcal{S}_n$ , this is given by

$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$ This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum. This is one plus [[St000234]].

Matching statistic: St000068
(load all 9 compositions to match this statistic)

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of minimal elements in a poset.

Matching statistic: St000084
(load all 6 compositions to match this statistic)

Mapping

Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000084: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of subtrees.

Matching statistic: St000286
(load all 5 compositions to match this statistic)

Mapping

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000286: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of connected components of the complement of a graph. The complement of a graph is the graph on the same vertex set with complementary edges.

The following 129 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000314The number of left-to-right-maxima of a permutation. St000382The first part of an integer composition. St000383The last part of an integer composition. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000759The smallest missing part in an integer partition. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000843The decomposition number of a perfect matching. St000991The number of right-to-left minima of a permutation. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001316The domatic number of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000439The position of the first down step of a Dyck path. St000546The number of global descents of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000010The length of the partition. St000015The number of peaks of a Dyck path. St000026The position of the first return of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000054The first entry of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000069The number of maximal elements of a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000335The difference of lower and upper interactions. St000363The number of minimal vertex covers of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000542The number of left-to-right-minima of a permutation. St000544The cop number of a graph. St000617The number of global maxima of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000822The Hadwiger number of the graph. St000883The number of longest increasing subsequences of a permutation. St000908The length of the shortest maximal antichain in a poset. St000916The packing number of a graph. St000971The smallest closer of a set partition. St000996The number of exclusive left-to-right maxima of a permutation. St001029The size of the core of a graph. St001068Number of torsionless simple modules 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. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001829The common independence number of a graph. St001963The tree-depth of a graph. St000051The size of the left subtree of a binary tree. St000053The number of valleys of the Dyck path. St000090The variation of a composition. St000133The "bounce" of a permutation. St000203The number of external nodes of a binary tree. St000272The treewidth of a graph. St000331The number of upper interactions of a Dyck path. St000362The size of a minimal vertex cover of a graph. St000504The cardinality of the first block of a set partition. St000536The pathwidth of a graph. St000738The first entry in the last row of a standard tableau. St000918The 2-limited packing number of a graph. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001169Number of simple modules with projective dimension at least two 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. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St000990The first ascent of a permutation. St000061The number of nodes on the left branch of a binary tree. St000654The first descent of a permutation. St000717The number of ordinal summands of a poset. St000989The number of final rises of a permutation. St000906The length of the shortest maximal chain in a poset. St000914The sum of the values of the Möbius function of a poset. St000925The number of topologically connected components of a set partition. St000993The multiplicity of the largest part of an integer partition. St000502The number of successions of a set partitions. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001812The biclique partition number of a graph. St001330The hat guessing number of a graph. St000648The number of 2-excedences of a permutation. St000924The number of topologically connected components of a perfect matching. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000260The radius of a connected graph. St000181The number of connected components of the Hasse diagram for the poset. 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. St000241The number of cyclical small excedances. St000315The number of isolated vertices of a graph. St000338The number of pixed 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. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000898The number of maximal entries in the last diagonal of the monotone triangle. St001889The size of the connectivity set of a signed permutation. 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. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000942The number of critical left to right maxima of the parking functions. St001904The length of the initial strictly increasing segment of a parking function. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.