- FindStat

Your data matches 14 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000148

Mapping

Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00186: Skew partitions —dominating partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000148: 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]
 => [[1,1],[]]
 => [1,1]
 => [2]
 => 0
[1,1,0,0]
 => [[2],[]]
 => [2]
 => [1,1]
 => 2
[1,0,1,0,1,0]
 => [[1,1,1],[]]
 => [1,1,1]
 => [3]
 => 1
[1,0,1,1,0,0]
 => [[2,1],[]]
 => [2,1]
 => [2,1]
 => 1
[1,1,0,0,1,0]
 => [[2,2],[1]]
 => [2,1]
 => [2,1]
 => 1
[1,1,0,1,0,0]
 => [[3],[]]
 => [3]
 => [1,1,1]
 => 3
[1,1,1,0,0,0]
 => [[2,2],[]]
 => [2,2]
 => [2,2]
 => 0
[1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => [1,1,1,1]
 => [4]
 => 0
[1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => [2,1,1]
 => [3,1]
 => 2
[1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [2,2]
 => [2,2]
 => 0
[1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => [3,1]
 => [2,1,1]
 => 2
[1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => [2,2,1]
 => [3,2]
 => 1
[1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [2,1,1]
 => [3,1]
 => 2
[1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [3,1]
 => [2,1,1]
 => 2
[1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [3,1]
 => [2,1,1]
 => 2
[1,1,0,1,0,1,0,0]
 => [[4],[]]
 => [4]
 => [1,1,1,1]
 => 4
[1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [3,2]
 => [2,2,1]
 => 1
[1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [2,2,1]
 => [3,2]
 => 1
[1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => [3,2]
 => [2,2,1]
 => 1
[1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => [2,2,2]
 => [3,3]
 => 2
[1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => [3,3]
 => [2,2,2]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1],[]]
 => [1,1,1,1,1]
 => [5]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1],[]]
 => [2,1,1,1]
 => [4,1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [2,2,1]
 => [3,2]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1],[]]
 => [3,1,1]
 => [3,1,1]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => [2,2,1,1]
 => [4,2]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [2,2,1]
 => [3,2]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [3,2]
 => [2,2,1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => [3,2]
 => [2,2,1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => [4,1]
 => [2,1,1,1]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => [3,3]
 => [2,2,2]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [2,2,2]
 => [3,3]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => [3,2,1]
 => [3,2,1]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => [2,2,2,1]
 => [4,3]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => [3,3,1]
 => [3,2,2]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [2,1,1,1]
 => [4,1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [3,1,1]
 => [3,1,1]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [3,2]
 => [2,2,1]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => [4,1]
 => [2,1,1,1]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [3,2,1]
 => [3,2,1]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [3,1,1]
 => [3,1,1]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => [4,1]
 => [2,1,1,1]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [4,1]
 => [2,1,1,1]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => [5]
 => [1,1,1,1,1]
 => 5
[1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => [4,2]
 => [2,2,1,1]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [3,2,1]
 => [3,2,1]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [[4,3],[1]]
 => [4,2]
 => [2,2,1,1]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => [3,2,2]
 => [3,3,1]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [4,3]
 => [2,2,2,1]
 => 1

Description

The number of odd parts of a partition.

Matching statistic: St000475

Mapping

Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts equal to 1 in a partition.

Matching statistic: St000445

Mapping

Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of rises of length 1 of a Dyck path.

Matching statistic: St000992

Mapping

Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00186: Skew partitions —dominating partition⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%

Values

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

Description

The alternating sum of the parts of an integer partition. For a partition

$\lambda = (\lambda_1,\ldots,\lambda_k)$ , this is

$\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$ .

Matching statistic: St001126
(load all 2 compositions to match this statistic)

Mapping

Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001126: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 87%●distinct values known / distinct values provided: 73%

Values

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

Description

Number of simple module that are 1-regular in the corresponding Nakayama algebra.

Matching statistic: St000247

Mapping

Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 73% ●values known / values provided: 87%●distinct values known / distinct values provided: 73%

Values

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

Description

The number of singleton blocks of a set partition.

Matching statistic: St000022
(load all 3 compositions to match this statistic)

Mapping

Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of fixed points of a permutation.

Matching statistic: St000895
(load all 3 compositions to match this statistic)

Mapping

Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 64%

Values

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

Description

The number of ones on the main diagonal of an alternating sign matrix.

Matching statistic: St000241
(load all 3 compositions to match this statistic)

Mapping

Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 64%

Values

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

Description

The number of cyclical small excedances. A cyclical small excedance is an index

$i$ such that

$\pi_i = i+1$ considered cyclically.

Matching statistic: St001189

Mapping

Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 64%

Values

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

Description

The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.

The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000894The trace of an alternating sign matrix. St000237The number of small exceedances. St000239The number of small weak excedances. St001903The number of fixed points of a parking function.