- FindStat

Your data matches 28 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000022
(load all 21 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of fixed points of a permutation.

Matching statistic: St000475
(load all 8 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00239: Permutations —Corteel⟶ 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] => [1] => [1] => [1]
 => 1
[1,2] => [1,2] => [1,2] => [1,1]
 => 2
[2,1] => [2,1] => [2,1] => [2]
 => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,1,1]
 => 3
[1,3,2] => [1,3,2] => [1,3,2] => [2,1]
 => 1
[2,1,3] => [2,1,3] => [2,1,3] => [2,1]
 => 1
[2,3,1] => [3,2,1] => [2,3,1] => [3]
 => 0
[3,1,2] => [3,2,1] => [2,3,1] => [3]
 => 0
[3,2,1] => [3,2,1] => [2,3,1] => [3]
 => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 4
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
 => 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
 => 2
[1,3,4,2] => [1,4,3,2] => [1,3,4,2] => [3,1]
 => 1
[1,4,2,3] => [1,4,3,2] => [1,3,4,2] => [3,1]
 => 1
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [3,1]
 => 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,1]
 => 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,2]
 => 0
[2,3,1,4] => [3,2,1,4] => [2,3,1,4] => [3,1]
 => 1
[2,3,4,1] => [4,2,3,1] => [2,3,4,1] => [4]
 => 0
[2,4,1,3] => [3,4,1,2] => [4,3,2,1] => [2,2]
 => 0
[2,4,3,1] => [4,3,2,1] => [3,4,1,2] => [2,2]
 => 0
[3,1,2,4] => [3,2,1,4] => [2,3,1,4] => [3,1]
 => 1
[3,1,4,2] => [4,2,3,1] => [2,3,4,1] => [4]
 => 0
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [3,1]
 => 1
[3,2,4,1] => [4,2,3,1] => [2,3,4,1] => [4]
 => 0
[3,4,1,2] => [4,3,2,1] => [3,4,1,2] => [2,2]
 => 0
[3,4,2,1] => [4,3,2,1] => [3,4,1,2] => [2,2]
 => 0
[4,1,2,3] => [4,2,3,1] => [2,3,4,1] => [4]
 => 0
[4,1,3,2] => [4,2,3,1] => [2,3,4,1] => [4]
 => 0
[4,2,1,3] => [4,3,2,1] => [3,4,1,2] => [2,2]
 => 0
[4,2,3,1] => [4,3,2,1] => [3,4,1,2] => [2,2]
 => 0
[4,3,1,2] => [4,3,2,1] => [3,4,1,2] => [2,2]
 => 0
[4,3,2,1] => [4,3,2,1] => [3,4,1,2] => [2,2]
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => 3
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
 => 3
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => [3,1,1]
 => 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => [3,1,1]
 => 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => [3,1,1]
 => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
 => 3
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
 => 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [3,1,1]
 => 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => [4,1]
 => 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,4,3,2] => [2,2,1]
 => 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,4,5,2,3] => [2,2,1]
 => 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,3,4,2,5] => [3,1,1]
 => 2
[1,4,2,5,3] => [1,5,3,4,2] => [1,3,4,5,2] => [4,1]
 => 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [3,1,1]
 => 2
[1,4,3,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => [4,1]
 => 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,4,5,2,3] => [2,2,1]
 => 1
[2,4,1,6,3,5,8,7,10,9] => [3,5,1,6,2,4,8,7,10,9] => [6,3,2,5,4,1,8,7,10,9] => ?
 => ? = 0
[2,4,1,6,3,8,5,7,10,9] => [3,5,1,7,2,8,4,6,10,9] => [8,3,2,5,4,7,6,1,10,9] => ?
 => ? = 0
[2,4,1,6,3,8,5,10,7,9] => [3,5,1,7,2,9,4,10,6,8] => [10,3,2,5,4,7,6,9,8,1] => ?
 => ? = 0
[2,1,4,3,6,5,8,10,7,9] => [2,1,4,3,6,5,9,10,7,8] => [2,1,4,3,6,5,10,9,8,7] => ?
 => ? = 0
[2,1,4,3,6,8,5,10,7,9] => [2,1,4,3,7,9,5,10,6,8] => [2,1,4,3,10,7,6,9,8,5] => ?
 => ? = 0
[2,1,4,6,3,8,5,10,7,9] => [2,1,5,7,3,9,4,10,6,8] => [2,1,10,5,4,7,6,9,8,3] => ?
 => ? = 0

Description

The number of parts equal to 1 in a partition.

Matching statistic: St000895
(load all 13 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 64% ●values known / values provided: 69%●distinct values known / distinct values provided: 64%

Values

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

Description

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

Matching statistic: St000011
(load all 3 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

Matching statistic: St000237
(load all 2 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of small exceedances. This is the number of indices $i$ such that $\pi_i=i+1$.

Matching statistic: St000214

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 91%

Values

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

Description

The number of adjacencies of a permutation. An adjacency of a permutation $\pi$ is an index $i$ such that $\pi(i)-1 = \pi(i+1)$. Adjacencies are also known as ''small descents''. This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].

Matching statistic: St000674
(load all 124 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 82%

Values

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

Description

The number of hills of a Dyck path. A hill is a peak with up step starting and down step ending at height zero.

Matching statistic: St000247
(load all 2 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 73%

Values

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

Description

The number of singleton blocks of a set partition.

Matching statistic: St001126

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001126: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 73%

Values

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

Description

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

Matching statistic: St001691
(load all 2 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001691: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 73%

Values

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

Description

The number of kings in a graph. A vertex of a graph is a king, if all its neighbours have smaller degree. In particular, an isolated vertex is a king.

The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000221The number of strong fixed points of a permutation. St000315The number of isolated vertices of a graph. St000117The number of centered tunnels of a Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000164The number of short pairs. St000241The number of cyclical small excedances. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001826The maximal number of leaves on a vertex of a graph. St001672The restrained domination number of a graph. St001479The number of bridges of a graph. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St000456The monochromatic index of a connected graph. St000894The trace of an alternating sign matrix. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001903The number of fixed points of a parking function. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset.