- FindStat

Your data matches 98 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000487
(load all 14 compositions to match this statistic)

Mapping

St000487: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the shortest cycle of a permutation.

Matching statistic: St000655
(load all 9 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the minimal rise of a Dyck path. For the length of a maximal rise, see [[St000444]].

Matching statistic: St001075
(load all 10 compositions to match this statistic)

Mapping

Mp00151: Permutations —to cycle type⟶ Set partitions
St001075: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The minimal size of a block of a set partition.

Matching statistic: St000657
(load all 10 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The smallest part of an integer composition.

Matching statistic: St000993
(load all 6 compositions to match this statistic)

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The multiplicity of the largest part of an integer partition.

Matching statistic: St001038
(load all 3 compositions to match this statistic)

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The minimal height of a column in the parallelogram polyomino associated with the Dyck path.

Matching statistic: St000025

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[2,1] => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
[1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 1
[2,1,3] => [2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 1
[2,3,1] => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[3,1,2] => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[3,2,1] => [2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 1
[1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,2,4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,3,2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,3,4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,4,2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,4,3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[2,1,3,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[2,1,4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,3,1,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[2,3,4,1] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
[2,4,1,3] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
[2,4,3,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[3,1,2,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[3,1,4,2] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
[3,2,1,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[3,2,4,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[3,4,1,2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,4,2,1] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
[4,1,2,3] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
[4,1,3,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[4,2,1,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[4,2,3,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[4,3,1,2] => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
[4,3,2,1] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,2,4,3,5] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,2,4,5,3] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,2,5,3,4] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,2,5,4,3] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,3,2,4,5] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,3,2,5,4] => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,3,4,2,5] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,3,4,5,2] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,3,5,2,4] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,3,5,4,2] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,4,2,3,5] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,4,2,5,3] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,4,3,2,5] => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,4,3,5,2] => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,4,5,2,3] => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,4,5,3,2] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 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: St000026
(load all 2 compositions to match this statistic)

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The position of the first return of a Dyck path.

Matching statistic: St000297

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,1]
 => [2]
 => 100 => 1
[2,1] => [2]
 => [1,1]
 => 110 => 2
[1,2,3] => [1,1,1]
 => [3]
 => 1000 => 1
[1,3,2] => [2,1]
 => [2,1]
 => 1010 => 1
[2,1,3] => [2,1]
 => [2,1]
 => 1010 => 1
[2,3,1] => [3]
 => [1,1,1]
 => 1110 => 3
[3,1,2] => [3]
 => [1,1,1]
 => 1110 => 3
[3,2,1] => [2,1]
 => [2,1]
 => 1010 => 1
[1,2,3,4] => [1,1,1,1]
 => [4]
 => 10000 => 1
[1,2,4,3] => [2,1,1]
 => [3,1]
 => 10010 => 1
[1,3,2,4] => [2,1,1]
 => [3,1]
 => 10010 => 1
[1,3,4,2] => [3,1]
 => [2,1,1]
 => 10110 => 1
[1,4,2,3] => [3,1]
 => [2,1,1]
 => 10110 => 1
[1,4,3,2] => [2,1,1]
 => [3,1]
 => 10010 => 1
[2,1,3,4] => [2,1,1]
 => [3,1]
 => 10010 => 1
[2,1,4,3] => [2,2]
 => [2,2]
 => 1100 => 2
[2,3,1,4] => [3,1]
 => [2,1,1]
 => 10110 => 1
[2,3,4,1] => [4]
 => [1,1,1,1]
 => 11110 => 4
[2,4,1,3] => [4]
 => [1,1,1,1]
 => 11110 => 4
[2,4,3,1] => [3,1]
 => [2,1,1]
 => 10110 => 1
[3,1,2,4] => [3,1]
 => [2,1,1]
 => 10110 => 1
[3,1,4,2] => [4]
 => [1,1,1,1]
 => 11110 => 4
[3,2,1,4] => [2,1,1]
 => [3,1]
 => 10010 => 1
[3,2,4,1] => [3,1]
 => [2,1,1]
 => 10110 => 1
[3,4,1,2] => [2,2]
 => [2,2]
 => 1100 => 2
[3,4,2,1] => [4]
 => [1,1,1,1]
 => 11110 => 4
[4,1,2,3] => [4]
 => [1,1,1,1]
 => 11110 => 4
[4,1,3,2] => [3,1]
 => [2,1,1]
 => 10110 => 1
[4,2,1,3] => [3,1]
 => [2,1,1]
 => 10110 => 1
[4,2,3,1] => [2,1,1]
 => [3,1]
 => 10010 => 1
[4,3,1,2] => [4]
 => [1,1,1,1]
 => 11110 => 4
[4,3,2,1] => [2,2]
 => [2,2]
 => 1100 => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => [5]
 => 100000 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [4,1]
 => 100010 => 1
[1,2,4,3,5] => [2,1,1,1]
 => [4,1]
 => 100010 => 1
[1,2,4,5,3] => [3,1,1]
 => [3,1,1]
 => 100110 => 1
[1,2,5,3,4] => [3,1,1]
 => [3,1,1]
 => 100110 => 1
[1,2,5,4,3] => [2,1,1,1]
 => [4,1]
 => 100010 => 1
[1,3,2,4,5] => [2,1,1,1]
 => [4,1]
 => 100010 => 1
[1,3,2,5,4] => [2,2,1]
 => [3,2]
 => 10100 => 1
[1,3,4,2,5] => [3,1,1]
 => [3,1,1]
 => 100110 => 1
[1,3,4,5,2] => [4,1]
 => [2,1,1,1]
 => 101110 => 1
[1,3,5,2,4] => [4,1]
 => [2,1,1,1]
 => 101110 => 1
[1,3,5,4,2] => [3,1,1]
 => [3,1,1]
 => 100110 => 1
[1,4,2,3,5] => [3,1,1]
 => [3,1,1]
 => 100110 => 1
[1,4,2,5,3] => [4,1]
 => [2,1,1,1]
 => 101110 => 1
[1,4,3,2,5] => [2,1,1,1]
 => [4,1]
 => 100010 => 1
[1,4,3,5,2] => [3,1,1]
 => [3,1,1]
 => 100110 => 1
[1,4,5,2,3] => [2,2,1]
 => [3,2]
 => 10100 => 1
[1,4,5,3,2] => [4,1]
 => [2,1,1,1]
 => 101110 => 1

Description

The number of leading ones in a binary word.

Matching statistic: St000314

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of left-to-right-maxima of a permutation. An integer

$\sigma_i$ in the one-line notation of a permutation

$\sigma$ is a '''left-to-right-maximum''' if there does not exist a

$j < i$ such that

$\sigma_j > \sigma_i$ . This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.

The following 88 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000326The position of the first one in a binary word after appending a 1 at the end. St000382The first part of an integer composition. St000383The last part of an integer composition. St000617The number of global maxima of a Dyck path. St000654The first descent of a permutation. St000733The row containing the largest entry of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St000210Minimum over maximum difference of elements in cycles. St000439The position of the first down step of a Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001810The number of fixed points of a permutation smaller than its largest moved point. St000990The first ascent 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. St000667The greatest common divisor of the parts of the partition. St001571The Cartan determinant of the integer partition. St000120The number of left tunnels of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001933The largest multiplicity of a part in an integer partition. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000640The rank of the largest boolean interval in a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000260The radius of a connected graph. St000259The diameter of a connected graph. St000706The product of the factorials of the multiplicities of an integer partition. St001890The maximum magnitude of the Möbius function of a poset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000090The variation of a composition. St000781The number of proper colouring schemes of a Ferrers diagram. St001432The order dimension of the partition. St001780The order of promotion on the set of standard tableaux of given shape. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St000668The least common multiple of the parts of the partition. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001877Number of indecomposable injective modules with projective dimension 2. St000456The monochromatic index of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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. 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. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001280The number of parts of an integer partition that are at least two. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000454The largest eigenvalue of a graph if it is integral. St001096The size of the overlap set of a permutation. St000633The size of the automorphism group of a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001884The number of borders of a binary word.