- FindStat

Your data matches 18 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000147

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest part of an integer partition.

Matching statistic: St000010

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000734

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 83% ●values known / values provided: 97%●distinct values known / distinct values provided: 83%

Values

[1,0]
 => [1] => [1]
 => [[1]]
 => 1
[1,0,1,0]
 => [1,2] => [1,1]
 => [[1],[2]]
 => 1
[1,1,0,0]
 => [2,1] => [2]
 => [[1,2]]
 => 2
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => [[1,2],[3]]
 => 2
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => [[1,2],[3]]
 => 2
[1,1,0,1,0,0]
 => [2,3,1] => [3]
 => [[1,2,3]]
 => 3
[1,1,1,0,0,0]
 => [3,2,1] => [2,1]
 => [[1,2],[3]]
 => 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1]
 => [[1,2,3],[4]]
 => 3
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1]
 => [[1,2,3],[4]]
 => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => [[1,2,3,4]]
 => 4
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => [[1,2,3],[4]]
 => 3
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => [[1,2,3],[4]]
 => 3
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [2,2]
 => [[1,2],[3,4]]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,2]
 => [[1,2,3],[4,5]]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2]
 => [[1,2,3],[4,5]]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5]
 => [[1,2,3,4,5]]
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [3,2]
 => [[1,2,3],[4,5]]
 => 3
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,10,9,8,7,6,5,4,3,2,1,12] => [2,2,2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11],[12]]
 => ? = 2
[]
 => [] => []
 => []
 => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,6,7,8,10,11,9] => [3,1,1,1,1,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,8,9,10,11,2] => [10,1]
 => [[1,2,3,4,5,6,7,8,9,10],[11]]
 => ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,8,9,10,11,1] => [11]
 => [[1,2,3,4,5,6,7,8,9,10,11]]
 => ? = 11
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,5,6,7,8,11,10,9] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,2,4,5,6,7,8,9,10,11,1] => [10,1]
 => [[1,2,3,4,5,6,7,8,9,10],[11]]
 => ? = 10
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,8,9,10,11,12,2] => [11,1]
 => [[1,2,3,4,5,6,7,8,9,10,11],[12]]
 => ? = 11
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => [11,10,3,4,5,6,7,9,8,2,1] => [2,2,2,1,1,1,1,1]
 => [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
 => ? = 2
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => [11,10,4,3,5,6,7,8,9,2,1] => [2,2,2,1,1,1,1,1]
 => [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
 => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [11,2,3,4,5,6,7,8,9,10,1] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7,8,9,10,11] => [3,1,1,1,1,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,4,5,6,7,8,10,9,11] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [2,3,4,5,6,7,8,9,10,1,11] => [10,1]
 => [[1,2,3,4,5,6,7,8,9,10],[11]]
 => ? = 10
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [11,3,4,5,6,7,8,9,10,2,1] => [9,2]
 => [[1,2,3,4,5,6,7,8,9],[10,11]]
 => ? = 9
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,1,0]
 => [9,8,7,6,5,4,3,2,1,10,11] => [2,2,2,2,1,1,1]
 => [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
 => ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [10,2,3,4,5,6,7,8,9,11,1] => [3,1,1,1,1,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 3
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [3,2,4,1,6,5,8,7,10,9,12,11] => [3,2,2,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
 => ? = 3
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,5,4,6,3,8,7,10,9,12,11] => [3,2,2,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
 => ? = 3
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [3,2,5,4,6,1,8,7,10,9,12,11] => [4,2,2,2,1,1]
 => [[1,2,3,4],[5,6],[7,8],[9,10],[11],[12]]
 => ? = 4
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,7,6,8,5,10,9,12,11] => [3,2,2,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
 => ? = 3
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,5,4,7,6,8,3,10,9,12,11] => [4,2,2,2,1,1]
 => [[1,2,3,4],[5,6],[7,8],[9,10],[11],[12]]
 => ? = 4
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [3,2,5,4,7,6,8,1,10,9,12,11] => [5,2,2,1,1,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10],[11],[12]]
 => ? = 5
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,9,8,10,7,12,11] => [3,2,2,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
 => ? = 3
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [3,2,5,4,6,1,9,8,10,7,12,11] => [4,3,2,1,1,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
 => ? = 4
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [2,1,4,3,7,6,9,8,10,5,12,11] => [4,2,2,2,1,1]
 => [[1,2,3,4],[5,6],[7,8],[9,10],[11],[12]]
 => ? = 4
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [3,2,4,1,7,6,9,8,10,5,12,11] => [4,3,2,1,1,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
 => ? = 4
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [2,1,5,4,7,6,9,8,10,3,12,11] => [5,2,2,1,1,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10],[11],[12]]
 => ? = 5
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [3,2,5,4,7,6,9,8,10,1,12,11] => [6,2,1,1,1,1]
 => [[1,2,3,4,5,6],[7,8],[9],[10],[11],[12]]
 => ? = 6
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,4,3,6,5,8,7,11,10,12,9] => [3,2,2,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
 => ? = 3
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [3,2,5,4,6,1,8,7,11,10,12,9] => [4,3,2,1,1,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
 => ? = 4
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [3,2,4,1,7,6,8,5,11,10,12,9] => [3,3,3,1,1,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10],[11],[12]]
 => ? = 3
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,5,4,7,6,8,3,11,10,12,9] => [4,3,2,1,1,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
 => ? = 4
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [3,2,5,4,7,6,8,1,11,10,12,9] => [5,3,1,1,1,1]
 => [[1,2,3,4,5],[6,7,8],[9],[10],[11],[12]]
 => ? = 5
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [2,1,4,3,6,5,9,8,11,10,12,7] => [4,2,2,2,1,1]
 => [[1,2,3,4],[5,6],[7,8],[9,10],[11],[12]]
 => ? = 4
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,4,1,6,5,9,8,11,10,12,7] => [4,3,2,1,1,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
 => ? = 4
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [2,1,5,4,6,3,9,8,11,10,12,7] => [4,3,2,1,1,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
 => ? = 4
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,5,4,6,1,9,8,11,10,12,7] => [4,4,1,1,1,1]
 => [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
 => ? = 4
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [2,1,4,3,7,6,9,8,11,10,12,5] => [5,2,2,1,1,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10],[11],[12]]
 => ? = 5
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,4,1,7,6,9,8,11,10,12,5] => [5,3,1,1,1,1]
 => [[1,2,3,4,5],[6,7,8],[9],[10],[11],[12]]
 => ? = 5
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [2,1,5,4,7,6,9,8,11,10,12,3] => [6,2,1,1,1,1]
 => [[1,2,3,4,5,6],[7,8],[9],[10],[11],[12]]
 => ? = 6
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,5,4,7,6,9,8,11,10,12,1] => [7,1,1,1,1,1]
 => [[1,2,3,4,5,6,7],[8],[9],[10],[11],[12]]
 => ? = 7
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [2,3,4,5,6,7,8,9,1,11,10] => [9,2]
 => [[1,2,3,4,5,6,7,8,9],[10,11]]
 => ? = 9
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,1,4,3,6,5,8,7,10,12,11,9] => [3,2,2,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
 => ? = 3
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [2,1,4,3,6,5,8,10,9,7,12,11] => [3,2,2,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
 => ? = 3
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,8,7,5,10,9,12,11] => [3,2,2,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
 => ? = 3
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,6,5,3,8,7,10,9,12,11] => [3,2,2,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
 => ? = 3

Description

The last entry in the first row of a standard tableau.

Matching statistic: St000676

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 87%●distinct values known / distinct values provided: 75%

Values

[1,0]
 => [1] => [1]
 => [1,0]
 => 1
[1,0,1,0]
 => [1,2] => [1,1]
 => [1,1,0,0]
 => 1
[1,1,0,0]
 => [2,1] => [2]
 => [1,0,1,0]
 => 2
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => [1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => [1,0,1,1,0,0]
 => 2
[1,1,0,1,0,0]
 => [2,3,1] => [3]
 => [1,0,1,0,1,0]
 => 3
[1,1,1,0,0,0]
 => [3,2,1] => [2,1]
 => [1,0,1,1,0,0]
 => 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => [1,1,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [2,2]
 => [1,1,1,0,0,0]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,10,9,8,7,6,5,4,3,2,1,12] => [2,2,2,2,2,1,1]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
 => ? = 2
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,10,9,8,5,6,7,4,3,2] => [2,2,2,1,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,9,8,4,5,6,7,3,2] => [2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,8,9,1] => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 9
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,6,7,9,8,1] => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,4,5,6,8,7,9,1] => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,4,5,6,9,7,8,1] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [2,3,4,5,7,6,8,9,1] => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [2,3,4,5,8,6,7,9,1] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [2,3,4,5,9,6,7,8,1] => [6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 6
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [2,3,4,6,5,7,8,9,1] => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [2,3,4,7,5,6,8,9,1] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7
[1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,4,3,5,6,7,8,9,1] => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [2,9,3,4,5,6,7,8,1] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,2,4,5,6,7,8,9,1] => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,7,6,9,8,1] => [5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 5
[1,1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [3,2,9,4,5,6,7,8,1] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [4,2,3,5,6,7,8,9,1] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7
[1,1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,9,6,7,8,1] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
 => [7,2,3,4,5,6,9,8,1] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [8,2,3,4,5,6,7,9,1] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [9,2,3,4,5,6,7,8,1] => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
 => [9,2,7,4,5,6,3,8,1] => [2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [9,3,2,4,5,6,7,8,1] => [2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0,0]
 => [9,4,3,2,5,6,7,8,1] => [2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8,9] => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,7,9,8] => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,6,8,9,7] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,4,5,7,8,9,6] => [4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,3,4,6,7,8,9,5] => [5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,8,9,4] => [6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 6
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,8,9,3] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,8,9,2] => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8,9,10] => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,7,8,10,9] => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,5,6,9,8,7] => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,6,7,9,10,8] => [3,1,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,4,5,8,7,9,6] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,4,5,6,8,9,10,7] => [4,1,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,3,4,7,6,8,9,5] => [4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 4
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,3,4,5,7,8,9,10,6] => [5,1,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,2,3,6,5,7,8,9,4] => [5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,3,4,6,7,8,9,10,5] => [6,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 6
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,2,5,4,6,7,8,9,3] => [6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,8,9,10,4] => [7,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 7
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,4,3,5,6,7,8,9,2] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,8,9,10,3] => [8,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 8
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,8,9,10,2] => [9,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 9
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,8,9,10,1] => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10

Description

The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength

$n$ with

$k$ up steps in odd positions and

$k$ returns to the main diagonal are counted by the binomial coefficient

$\binom{n-1}{k-1}$ [3,4].

Matching statistic: St001039

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 87%●distinct values known / distinct values provided: 67%

Values

[1,0]
 => [1] => [1]
 => [1,0]
 => ? = 1
[1,0,1,0]
 => [1,2] => [1,1]
 => [1,1,0,0]
 => 1
[1,1,0,0]
 => [2,1] => [2]
 => [1,0,1,0]
 => 2
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => [1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => [1,0,1,1,0,0]
 => 2
[1,1,0,1,0,0]
 => [2,3,1] => [3]
 => [1,0,1,0,1,0]
 => 3
[1,1,1,0,0,0]
 => [3,2,1] => [2,1]
 => [1,0,1,1,0,0]
 => 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => [1,1,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [2,2]
 => [1,1,1,0,0,0]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,10,9,8,7,6,5,4,3,2,1,12] => [2,2,2,2,2,1,1]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
 => ? = 2
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,10,9,8,5,6,7,4,3,2] => [2,2,2,1,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,9,8,4,5,6,7,3,2] => [2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[]
 => [] => []
 => []
 => ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,8,9,1] => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 9
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,6,7,9,8,1] => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,4,5,6,8,7,9,1] => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,4,5,6,9,7,8,1] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [2,3,4,5,7,6,8,9,1] => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [2,3,4,5,8,6,7,9,1] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [2,3,4,5,9,6,7,8,1] => [6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 6
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [2,3,4,6,5,7,8,9,1] => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [2,3,4,7,5,6,8,9,1] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7
[1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,4,3,5,6,7,8,9,1] => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [2,9,3,4,5,6,7,8,1] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,2,4,5,6,7,8,9,1] => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,7,6,9,8,1] => [5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 5
[1,1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [3,2,9,4,5,6,7,8,1] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [4,2,3,5,6,7,8,9,1] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7
[1,1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,9,6,7,8,1] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
 => [7,2,3,4,5,6,9,8,1] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [8,2,3,4,5,6,7,9,1] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [9,2,3,4,5,6,7,8,1] => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
 => [9,2,7,4,5,6,3,8,1] => [2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [9,3,2,4,5,6,7,8,1] => [2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0,0]
 => [9,4,3,2,5,6,7,8,1] => [2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8,9] => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,7,9,8] => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,6,8,9,7] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,4,5,7,8,9,6] => [4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,3,4,6,7,8,9,5] => [5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,8,9,4] => [6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 6
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,8,9,3] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,8,9,2] => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8,9,10] => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,7,8,10,9] => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,5,6,9,8,7] => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,6,7,9,10,8] => [3,1,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,4,5,8,7,9,6] => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,4,5,6,8,9,10,7] => [4,1,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,3,4,7,6,8,9,5] => [4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 4
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,3,4,5,7,8,9,10,6] => [5,1,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,2,3,6,5,7,8,9,4] => [5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,3,4,6,7,8,9,10,5] => [6,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 6
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,2,5,4,6,7,8,9,3] => [6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 6
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,8,9,10,4] => [7,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 7
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,4,3,5,6,7,8,9,2] => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,8,9,10,3] => [8,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 8

Description

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

Matching statistic: St000392

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 92%

Values

[1,0]
 => [1] => [1] =>  => ? = 1 - 1
[1,0,1,0]
 => [1,2] => [1,2] => 0 => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => [2,1] => 1 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 00 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 01 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 10 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [3,2,1] => 11 => 2 = 3 - 1
[1,1,1,0,0,0]
 => [3,2,1] => [2,3,1] => 01 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => 001 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => 010 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => 011 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,3,4,2] => 001 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => 100 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => 101 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,2,1,4] => 110 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,3,2,1] => 111 => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,4,2,1] => 011 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,3,1,4] => 010 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,4,3,1] => 011 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [2,3,4,1] => 001 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [3,2,4,1] => 101 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,4,3] => 0011 => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,4,5,3] => 0001 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,4,3,2,5] => 0110 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,4,3,2] => 0111 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,4,5,3,2] => 0011 => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,3,4,2,5] => 0010 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,3,5,4,2] => 0011 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [1,3,4,5,2] => 0001 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,4,3,5,2] => 0101 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => 1001 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => 1010 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => 1011 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,4,5,3] => 1001 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,2,1,4,5] => 1100 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2,1,5,4] => 1101 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,3,2,1,5] => 1110 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,4,3,2,1] => 1111 => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [4,5,3,2,1] => 0111 => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,4,2,1,5] => 0110 => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,5,4,2,1] => 0111 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [3,4,5,2,1] => 0011 => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [4,3,5,2,1] => 1011 => 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [2,3,1,4,5] => 0100 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,2,4,3,6,7,5,8] => [1,2,4,3,7,6,5,8] => ? => ? = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,2,4,5,3,7,6,8] => [1,2,5,4,3,7,6,8] => ? => ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,2,4,5,6,3,8,7] => [1,2,6,5,4,3,8,7] => ? => ? = 4 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,2,4,6,5,8,7,3] => [1,2,5,7,8,6,4,3] => ? => ? = 4 - 1
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,2,4,8,7,6,5,3] => [1,2,6,7,5,8,4,3] => ? => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,2,5,4,3,6,7,8] => [1,2,4,5,3,6,7,8] => ? => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,2,5,4,3,6,8,7] => [1,2,4,5,3,6,8,7] => ? => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,2,5,4,6,7,3,8] => [1,2,4,7,6,5,3,8] => ? => ? = 4 - 1
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,2,8,4,7,6,5,3] => [1,2,4,6,7,5,8,3] => ? => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,2,6,5,4,3,7,8] => [1,2,5,4,6,3,7,8] => ? => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,2,6,5,4,8,7,3] => [1,2,5,4,7,8,6,3] => ? => ? = 3 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,2,7,6,5,4,8,3] => [1,2,5,6,4,8,7,3] => ? => ? = 3 - 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,8,7,6,5,4,3] => [1,2,6,5,7,4,8,3] => ? => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => [1,3,2,4,6,8,7,5] => [1,3,2,4,7,8,6,5] => ? => ? = 3 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,3,2,4,7,6,5,8] => [1,3,2,4,6,7,5,8] => ? => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => [1,3,4,2,6,5,8,7] => [1,4,3,2,6,5,8,7] => ? => ? = 3 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [1,3,4,2,6,8,7,5] => [1,4,3,2,7,8,6,5] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,1,0,0]
 => [1,3,6,4,5,2,8,7] => [1,4,5,6,3,2,8,7] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
 => [1,3,6,4,5,7,2,8] => [1,4,5,7,6,3,2,8] => ? => ? = 4 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [1,3,7,4,5,6,8,2] => [1,4,5,6,8,7,3,2] => ? => ? = 4 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => [1,3,8,4,6,5,7,2] => [1,4,6,5,7,8,3,2] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => [1,3,8,5,4,7,6,2] => [1,5,4,7,6,8,3,2] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [1,3,8,5,7,6,4,2] => [1,6,7,5,4,8,3,2] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => [1,3,8,6,5,7,4,2] => [1,5,7,6,4,8,3,2] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [1,3,8,7,5,6,4,2] => [1,5,6,7,4,8,3,2] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,3,8,7,6,5,4,2] => [1,6,5,7,4,8,3,2] => ? => ? = 3 - 1
[1,0,1,1,1,0,0,0,1,1,0,1,0,0,1,0]
 => [1,4,3,2,6,7,5,8] => [1,3,4,2,7,6,5,8] => ? => ? = 3 - 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => [1,4,3,5,6,2,7,8] => [1,3,6,5,4,2,7,8] => ? => ? = 4 - 1
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [1,4,3,5,8,7,6,2] => [1,3,7,6,8,5,4,2] => ? => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
 => [1,5,3,4,6,2,8,7] => [1,3,4,6,5,2,8,7] => ? => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => [1,5,3,4,6,7,2,8] => [1,3,4,7,6,5,2,8] => ? => ? = 4 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [1,8,3,5,6,4,7,2] => [1,3,6,5,4,7,8,2] => ? => ? = 3 - 1
[1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [1,8,3,5,6,7,4,2] => [1,3,7,6,5,4,8,2] => ? => ? = 4 - 1
[1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [1,8,3,5,7,6,4,2] => [1,3,6,7,5,4,8,2] => ? => ? = 3 - 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => [1,7,3,6,5,4,8,2] => [1,3,5,6,4,8,7,2] => ? => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,5,4,3,2,6,7,8] => [1,4,3,5,2,6,7,8] => ? => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => [1,5,4,3,6,7,2,8] => [1,4,3,7,6,5,2,8] => ? => ? = 4 - 1
[1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => [1,8,4,3,5,7,6,2] => [1,4,3,5,7,6,8,2] => ? => ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,1,0,0]
 => [1,6,4,5,3,2,8,7] => [1,5,4,3,6,2,8,7] => ? => ? = 3 - 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,7,4,6,5,3,2,8] => [1,5,6,4,3,7,2,8] => ? => ? = 3 - 1
[1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [1,8,4,6,5,7,3,2] => [1,5,7,6,4,3,8,2] => ? => ? = 4 - 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,6,5,4,3,2,8,7] => [1,4,5,3,6,2,8,7] => ? => ? = 2 - 1
[1,0,1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [1,8,5,4,3,7,6,2] => [1,4,5,3,7,6,8,2] => ? => ? = 2 - 1
[1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [1,8,5,4,6,3,7,2] => [1,4,6,5,3,7,8,2] => ? => ? = 3 - 1
[1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [1,8,5,4,6,7,3,2] => [1,4,7,6,5,3,8,2] => ? => ? = 4 - 1
[1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [1,8,6,4,5,3,7,2] => [1,4,5,6,3,7,8,2] => ? => ? = 2 - 1
[1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,8,6,4,5,7,3,2] => [1,4,5,7,6,3,8,2] => ? => ? = 3 - 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [1,7,6,5,4,3,8,2] => [1,5,4,6,3,8,7,2] => ? => ? = 3 - 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,8,6,5,4,7,3,2] => [1,5,4,7,6,3,8,2] => ? => ? = 3 - 1

Description

The length of the longest run of ones in a binary word.

Matching statistic: St001291

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 70%●distinct values known / distinct values provided: 42%

Values

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

Description

The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. Let

$A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of

$D(A) \otimes D(A)$ , where

$D(A)$ is the natural dual of

$A$ .

Matching statistic: St000381

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 83%

Values

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

Description

The largest part of an integer composition.

Matching statistic: St001062
(load all 4 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00176: Set partitions —rotate decreasing⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%

Values

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

Description

The maximal size of a block of a set partition.

Matching statistic: St000451

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 67%

Values

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

Description

The length of the longest pattern of the form k 1 2...(k-1).

The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000308The height of the tree associated to a permutation. St000485The length of the longest cycle of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001330The hat guessing number of a graph. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000455The second largest eigenvalue of a graph if it is integral. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ .