Your data matches 12 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000022
(load all 33 compositions to match this statistic)
Mapping
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 2
[2,1] => 0
[1,2,3] => 3
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 4
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 2
[2,1,3,4] => 2
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 0
[3,2,1,4] => 2
[3,2,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 2
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 5
[1,2,3,5,4] => 3
[1,2,4,3,5] => 3
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 3
[1,3,2,4,5] => 3
[1,3,2,5,4] => 1
[1,3,4,2,5] => 2
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 2
[1,4,2,3,5] => 2
[1,4,2,5,3] => 1
[1,4,3,2,5] => 3
[1,4,3,5,2] => 2
[1,4,5,2,3] => 1
[1,4,5,3,2] => 1
Description
The number of fixed points of a permutation.
Matching statistic: St000241
(load all 20 compositions to match this statistic)
Mapping
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => 2
[2,1] => [1,2] => 0
[1,2,3] => [2,3,1] => 3
[1,3,2] => [3,2,1] => 1
[2,1,3] => [1,3,2] => 1
[2,3,1] => [1,2,3] => 0
[3,1,2] => [3,1,2] => 0
[3,2,1] => [2,1,3] => 1
[1,2,3,4] => [2,3,4,1] => 4
[1,2,4,3] => [2,4,3,1] => 2
[1,3,2,4] => [3,2,4,1] => 2
[1,3,4,2] => [4,2,3,1] => 1
[1,4,2,3] => [3,4,2,1] => 1
[1,4,3,2] => [4,3,2,1] => 2
[2,1,3,4] => [1,3,4,2] => 2
[2,1,4,3] => [1,4,3,2] => 0
[2,3,1,4] => [1,2,4,3] => 1
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,4,2,3] => 0
[2,4,3,1] => [1,3,2,4] => 1
[3,1,2,4] => [3,1,4,2] => 1
[3,1,4,2] => [4,1,3,2] => 0
[3,2,1,4] => [2,1,4,3] => 2
[3,2,4,1] => [2,1,3,4] => 1
[3,4,1,2] => [4,1,2,3] => 0
[3,4,2,1] => [3,1,2,4] => 0
[4,1,2,3] => [3,4,1,2] => 0
[4,1,3,2] => [4,3,1,2] => 1
[4,2,1,3] => [2,4,1,3] => 1
[4,2,3,1] => [2,3,1,4] => 2
[4,3,1,2] => [4,2,1,3] => 0
[4,3,2,1] => [3,2,1,4] => 0
[1,2,3,4,5] => [2,3,4,5,1] => 5
[1,2,3,5,4] => [2,3,5,4,1] => 3
[1,2,4,3,5] => [2,4,3,5,1] => 3
[1,2,4,5,3] => [2,5,3,4,1] => 2
[1,2,5,3,4] => [2,4,5,3,1] => 2
[1,2,5,4,3] => [2,5,4,3,1] => 3
[1,3,2,4,5] => [3,2,4,5,1] => 3
[1,3,2,5,4] => [3,2,5,4,1] => 1
[1,3,4,2,5] => [4,2,3,5,1] => 2
[1,3,4,5,2] => [5,2,3,4,1] => 1
[1,3,5,2,4] => [4,2,5,3,1] => 1
[1,3,5,4,2] => [5,2,4,3,1] => 2
[1,4,2,3,5] => [3,4,2,5,1] => 2
[1,4,2,5,3] => [3,5,2,4,1] => 1
[1,4,3,2,5] => [4,3,2,5,1] => 3
[1,4,3,5,2] => [5,3,2,4,1] => 2
[1,4,5,2,3] => [4,5,2,3,1] => 1
[1,4,5,3,2] => [5,4,2,3,1] => 1
Description
The number of cyclical small excedances. A cyclical small excedance is an index $i$ such that $\pi_i = i+1$ considered cyclically.
Matching statistic: St000247
(load all 40 compositions to match this statistic)
Mapping
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => {{1},{2}}
 => 2
[2,1] => {{1,2}}
 => 0
[1,2,3] => {{1},{2},{3}}
 => 3
[1,3,2] => {{1},{2,3}}
 => 1
[2,1,3] => {{1,2},{3}}
 => 1
[2,3,1] => {{1,2,3}}
 => 0
[3,1,2] => {{1,2,3}}
 => 0
[3,2,1] => {{1,3},{2}}
 => 1
[1,2,3,4] => {{1},{2},{3},{4}}
 => 4
[1,2,4,3] => {{1},{2},{3,4}}
 => 2
[1,3,2,4] => {{1},{2,3},{4}}
 => 2
[1,3,4,2] => {{1},{2,3,4}}
 => 1
[1,4,2,3] => {{1},{2,3,4}}
 => 1
[1,4,3,2] => {{1},{2,4},{3}}
 => 2
[2,1,3,4] => {{1,2},{3},{4}}
 => 2
[2,1,4,3] => {{1,2},{3,4}}
 => 0
[2,3,1,4] => {{1,2,3},{4}}
 => 1
[2,3,4,1] => {{1,2,3,4}}
 => 0
[2,4,1,3] => {{1,2,3,4}}
 => 0
[2,4,3,1] => {{1,2,4},{3}}
 => 1
[3,1,2,4] => {{1,2,3},{4}}
 => 1
[3,1,4,2] => {{1,2,3,4}}
 => 0
[3,2,1,4] => {{1,3},{2},{4}}
 => 2
[3,2,4,1] => {{1,3,4},{2}}
 => 1
[3,4,1,2] => {{1,3},{2,4}}
 => 0
[3,4,2,1] => {{1,2,3,4}}
 => 0
[4,1,2,3] => {{1,2,3,4}}
 => 0
[4,1,3,2] => {{1,2,4},{3}}
 => 1
[4,2,1,3] => {{1,3,4},{2}}
 => 1
[4,2,3,1] => {{1,4},{2},{3}}
 => 2
[4,3,1,2] => {{1,2,3,4}}
 => 0
[4,3,2,1] => {{1,4},{2,3}}
 => 0
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 5
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 3
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 3
[1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 2
[1,2,5,3,4] => {{1},{2},{3,4,5}}
 => 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
 => 3
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => 3
[1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
 => 2
[1,3,4,5,2] => {{1},{2,3,4,5}}
 => 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
 => 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
 => 2
[1,4,2,3,5] => {{1},{2,3,4},{5}}
 => 2
[1,4,2,5,3] => {{1},{2,3,4,5}}
 => 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
 => 3
[1,4,3,5,2] => {{1},{2,4,5},{3}}
 => 2
[1,4,5,2,3] => {{1},{2,4},{3,5}}
 => 1
[1,4,5,3,2] => {{1},{2,3,4,5}}
 => 1
Description
The number of singleton blocks of a set partition.
Matching statistic: St000475
(load all 4 compositions to match this statistic)
Mapping
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
[1,2] => [1,1]
 => 2
[2,1] => [2]
 => 0
[1,2,3] => [1,1,1]
 => 3
[1,3,2] => [2,1]
 => 1
[2,1,3] => [2,1]
 => 1
[2,3,1] => [3]
 => 0
[3,1,2] => [3]
 => 0
[3,2,1] => [2,1]
 => 1
[1,2,3,4] => [1,1,1,1]
 => 4
[1,2,4,3] => [2,1,1]
 => 2
[1,3,2,4] => [2,1,1]
 => 2
[1,3,4,2] => [3,1]
 => 1
[1,4,2,3] => [3,1]
 => 1
[1,4,3,2] => [2,1,1]
 => 2
[2,1,3,4] => [2,1,1]
 => 2
[2,1,4,3] => [2,2]
 => 0
[2,3,1,4] => [3,1]
 => 1
[2,3,4,1] => [4]
 => 0
[2,4,1,3] => [4]
 => 0
[2,4,3,1] => [3,1]
 => 1
[3,1,2,4] => [3,1]
 => 1
[3,1,4,2] => [4]
 => 0
[3,2,1,4] => [2,1,1]
 => 2
[3,2,4,1] => [3,1]
 => 1
[3,4,1,2] => [2,2]
 => 0
[3,4,2,1] => [4]
 => 0
[4,1,2,3] => [4]
 => 0
[4,1,3,2] => [3,1]
 => 1
[4,2,1,3] => [3,1]
 => 1
[4,2,3,1] => [2,1,1]
 => 2
[4,3,1,2] => [4]
 => 0
[4,3,2,1] => [2,2]
 => 0
[1,2,3,4,5] => [1,1,1,1,1]
 => 5
[1,2,3,5,4] => [2,1,1,1]
 => 3
[1,2,4,3,5] => [2,1,1,1]
 => 3
[1,2,4,5,3] => [3,1,1]
 => 2
[1,2,5,3,4] => [3,1,1]
 => 2
[1,2,5,4,3] => [2,1,1,1]
 => 3
[1,3,2,4,5] => [2,1,1,1]
 => 3
[1,3,2,5,4] => [2,2,1]
 => 1
[1,3,4,2,5] => [3,1,1]
 => 2
[1,3,4,5,2] => [4,1]
 => 1
[1,3,5,2,4] => [4,1]
 => 1
[1,3,5,4,2] => [3,1,1]
 => 2
[1,4,2,3,5] => [3,1,1]
 => 2
[1,4,2,5,3] => [4,1]
 => 1
[1,4,3,2,5] => [2,1,1,1]
 => 3
[1,4,3,5,2] => [3,1,1]
 => 2
[1,4,5,2,3] => [2,2,1]
 => 1
[1,4,5,3,2] => [4,1]
 => 1
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000894
(load all 11 compositions to match this statistic)
Mapping
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [[1,0],[0,1]]
 => 2
[2,1] => [[0,1],[1,0]]
 => 0
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
 => 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => 1
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => 0
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
 => 0
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => 1
[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] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => 2
[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] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => 1
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => 1
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => 2
[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] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => 0
[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] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => 0
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => 1
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => 1
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => 0
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => 2
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => 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] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => 0
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => 0
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => 1
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => 1
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => 2
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
 => 0
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 0
[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] => [[1,0,0,0,0],[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] => [[1,0,0,0,0],[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] => [[1,0,0,0,0],[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] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
 => 2
[1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => 3
[1,3,2,4,5] => [[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] => [[1,0,0,0,0],[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] => [[1,0,0,0,0],[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] => [[1,0,0,0,0],[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] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
 => 1
[1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
 => 2
[1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => 2
[1,4,2,5,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => 1
[1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => 3
[1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
 => 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
[1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
 => 1
Description
The trace of an alternating sign matrix.
Matching statistic: St000895
(load all 11 compositions to match this statistic)
Mapping
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [[1,0],[0,1]]
 => 2
[2,1] => [[0,1],[1,0]]
 => 0
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
 => 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => 1
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => 0
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
 => 0
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => 1
[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] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => 2
[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] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => 1
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => 1
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => 2
[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] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => 0
[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] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => 0
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => 1
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => 1
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => 0
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => 2
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => 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] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => 0
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => 0
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => 1
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => 1
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => 2
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
 => 0
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 0
[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] => [[1,0,0,0,0],[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] => [[1,0,0,0,0],[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] => [[1,0,0,0,0],[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] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
 => 2
[1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => 3
[1,3,2,4,5] => [[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] => [[1,0,0,0,0],[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] => [[1,0,0,0,0],[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] => [[1,0,0,0,0],[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] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
 => 1
[1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
 => 2
[1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => 2
[1,4,2,5,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
 => 1
[1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => 3
[1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
 => 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
[1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
 => 1
Description
The number of ones on the main diagonal of an alternating sign matrix.
Matching statistic: St000248
(load all 8 compositions to match this statistic)
Mapping
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => {{1},{2}}
 => {{1,2}}
 => 2
[2,1] => {{1,2}}
 => {{1},{2}}
 => 0
[1,2,3] => {{1},{2},{3}}
 => {{1,2,3}}
 => 3
[1,3,2] => {{1},{2,3}}
 => {{1,3},{2}}
 => 1
[2,1,3] => {{1,2},{3}}
 => {{1,2},{3}}
 => 1
[2,3,1] => {{1,2,3}}
 => {{1},{2},{3}}
 => 0
[3,1,2] => {{1,2,3}}
 => {{1},{2},{3}}
 => 0
[3,2,1] => {{1,3},{2}}
 => {{1},{2,3}}
 => 1
[1,2,3,4] => {{1},{2},{3},{4}}
 => {{1,2,3,4}}
 => 4
[1,2,4,3] => {{1},{2},{3,4}}
 => {{1,3,4},{2}}
 => 2
[1,3,2,4] => {{1},{2,3},{4}}
 => {{1,2,4},{3}}
 => 2
[1,3,4,2] => {{1},{2,3,4}}
 => {{1,4},{2},{3}}
 => 1
[1,4,2,3] => {{1},{2,3,4}}
 => {{1,4},{2},{3}}
 => 1
[1,4,3,2] => {{1},{2,4},{3}}
 => {{1,4},{2,3}}
 => 2
[2,1,3,4] => {{1,2},{3},{4}}
 => {{1,2,3},{4}}
 => 2
[2,1,4,3] => {{1,2},{3,4}}
 => {{1,3},{2},{4}}
 => 0
[2,3,1,4] => {{1,2,3},{4}}
 => {{1,2},{3},{4}}
 => 1
[2,3,4,1] => {{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => 0
[2,4,1,3] => {{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => 0
[2,4,3,1] => {{1,2,4},{3}}
 => {{1},{2,3},{4}}
 => 1
[3,1,2,4] => {{1,2,3},{4}}
 => {{1,2},{3},{4}}
 => 1
[3,1,4,2] => {{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => 0
[3,2,1,4] => {{1,3},{2},{4}}
 => {{1,2},{3,4}}
 => 2
[3,2,4,1] => {{1,3,4},{2}}
 => {{1},{2},{3,4}}
 => 1
[3,4,1,2] => {{1,3},{2,4}}
 => {{1,3},{2,4}}
 => 0
[3,4,2,1] => {{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => 0
[4,1,2,3] => {{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => 0
[4,1,3,2] => {{1,2,4},{3}}
 => {{1},{2,3},{4}}
 => 1
[4,2,1,3] => {{1,3,4},{2}}
 => {{1},{2},{3,4}}
 => 1
[4,2,3,1] => {{1,4},{2},{3}}
 => {{1},{2,3,4}}
 => 2
[4,3,1,2] => {{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => 0
[4,3,2,1] => {{1,4},{2,3}}
 => {{1},{2,4},{3}}
 => 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},{4,5}}
 => {{1,3,4,5},{2}}
 => 3
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => {{1,2,4,5},{3}}
 => 3
[1,2,4,5,3] => {{1},{2},{3,4,5}}
 => {{1,4,5},{2},{3}}
 => 2
[1,2,5,3,4] => {{1},{2},{3,4,5}}
 => {{1,4,5},{2},{3}}
 => 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
 => {{1,4,5},{2,3}}
 => 3
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => {{1,2,3,5},{4}}
 => 3
[1,3,2,5,4] => {{1},{2,3},{4,5}}
 => {{1,3,5},{2},{4}}
 => 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
 => {{1,2,5},{3},{4}}
 => 2
[1,3,4,5,2] => {{1},{2,3,4,5}}
 => {{1,5},{2},{3},{4}}
 => 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
 => {{1,5},{2},{3},{4}}
 => 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
 => {{1,5},{2,3},{4}}
 => 2
[1,4,2,3,5] => {{1},{2,3,4},{5}}
 => {{1,2,5},{3},{4}}
 => 2
[1,4,2,5,3] => {{1},{2,3,4,5}}
 => {{1,5},{2},{3},{4}}
 => 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
 => {{1,2,5},{3,4}}
 => 3
[1,4,3,5,2] => {{1},{2,4,5},{3}}
 => {{1,5},{2},{3,4}}
 => 2
[1,4,5,2,3] => {{1},{2,4},{3,5}}
 => {{1,3,5},{2,4}}
 => 1
[1,4,5,3,2] => {{1},{2,3,4,5}}
 => {{1,5},{2},{3},{4}}
 => 1
Description
The number of anti-singletons of a set partition. An anti-singleton of a set partition $S$ is an index $i$ such that $i$ and $i+1$ (considered cyclically) are both in the same block of $S$. For noncrossing set partitions, this is also the number of singletons of the image of $S$ under the Kreweras complement.
Matching statistic: St000215
(load all 2 compositions to match this statistic)
Mapping
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000215: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [2,1] => [2,1] => 2
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [2,3,1] => [3,1,2] => [3,2,1] => 3
[1,3,2] => [3,2,1] => [3,2,1] => [2,3,1] => 1
[2,1,3] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [3,1,2] => [2,3,1] => [3,1,2] => 0
[3,2,1] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,2,3,4] => [2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 4
[1,2,4,3] => [2,4,3,1] => [4,1,3,2] => [3,4,2,1] => 2
[1,3,2,4] => [3,2,4,1] => [4,2,1,3] => [2,4,3,1] => 2
[1,3,4,2] => [4,2,3,1] => [4,2,3,1] => [2,3,4,1] => 1
[1,4,2,3] => [3,4,2,1] => [4,3,1,2] => [4,2,3,1] => 1
[1,4,3,2] => [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 2
[2,1,3,4] => [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 2
[2,1,4,3] => [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
[2,3,1,4] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 0
[2,4,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => [4,3,1,2] => 1
[3,1,4,2] => [4,1,3,2] => [2,4,3,1] => [3,4,1,2] => 0
[3,2,1,4] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,2,4,1] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[3,4,1,2] => [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 0
[3,4,2,1] => [3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 0
[4,1,2,3] => [3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 0
[4,1,3,2] => [4,3,1,2] => [3,4,2,1] => [4,1,3,2] => 1
[4,2,1,3] => [2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 1
[4,2,3,1] => [2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 2
[4,3,1,2] => [4,2,1,3] => [3,2,4,1] => [2,4,1,3] => 0
[4,3,2,1] => [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => [5,4,3,2,1] => 5
[1,2,3,5,4] => [2,3,5,4,1] => [5,1,2,4,3] => [4,5,3,2,1] => 3
[1,2,4,3,5] => [2,4,3,5,1] => [5,1,3,2,4] => [3,5,4,2,1] => 3
[1,2,4,5,3] => [2,5,3,4,1] => [5,1,3,4,2] => [3,4,5,2,1] => 2
[1,2,5,3,4] => [2,4,5,3,1] => [5,1,4,2,3] => [5,3,4,2,1] => 2
[1,2,5,4,3] => [2,5,4,3,1] => [5,1,4,3,2] => [4,3,5,2,1] => 3
[1,3,2,4,5] => [3,2,4,5,1] => [5,2,1,3,4] => [2,5,4,3,1] => 3
[1,3,2,5,4] => [3,2,5,4,1] => [5,2,1,4,3] => [2,4,5,3,1] => 1
[1,3,4,2,5] => [4,2,3,5,1] => [5,2,3,1,4] => [2,3,5,4,1] => 2
[1,3,4,5,2] => [5,2,3,4,1] => [5,2,3,4,1] => [2,3,4,5,1] => 1
[1,3,5,2,4] => [4,2,5,3,1] => [5,2,4,1,3] => [2,5,3,4,1] => 1
[1,3,5,4,2] => [5,2,4,3,1] => [5,2,4,3,1] => [2,4,3,5,1] => 2
[1,4,2,3,5] => [3,4,2,5,1] => [5,3,1,2,4] => [5,4,2,3,1] => 2
[1,4,2,5,3] => [3,5,2,4,1] => [5,3,1,4,2] => [4,5,2,3,1] => 1
[1,4,3,2,5] => [4,3,2,5,1] => [5,3,2,1,4] => [3,2,5,4,1] => 3
[1,4,3,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => [3,2,4,5,1] => 2
[1,4,5,2,3] => [4,5,2,3,1] => [5,3,4,1,2] => [5,2,3,4,1] => 1
[1,4,5,3,2] => [5,4,2,3,1] => [5,3,4,2,1] => [4,2,3,5,1] => 1
Description
The number of adjacencies of a permutation, zero appended. An adjacency is a descent of the form $(e+1,e)$ in the word corresponding to the permutation in one-line notation. This statistic, $\operatorname{adj_0}$, counts adjacencies in the word with a zero appended. $(\operatorname{adj_0}, \operatorname{des})$ and $(\operatorname{fix}, \operatorname{exc})$ are equidistributed, see [1].
Matching statistic: St000445
(load all 3 compositions to match this statistic)
Mapping
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[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] => [3,1,2] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 0
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[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] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[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] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[2,4,3,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[3,1,4,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[3,2,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[3,4,2,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[4,1,3,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[4,3,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 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] => [1,2,5,3,4] => [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] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
[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] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,3,5,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,3,5,4,2] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[1,4,2,3,5] => [1,4,2,3,5] => [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] => [1,5,4,2,3] => [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] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3
[1,4,3,5,2] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,4,5,2,3] => [1,5,2,4,3] => [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] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000674
(load all 4 compositions to match this statistic)
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[2,1] => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,2,3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3
[1,3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[2,1,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[2,3,1] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[3,1,2] => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[3,2,1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,2,3,4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,2,4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[1,3,2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[1,3,4,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,4,2,3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,4,3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[2,1,3,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[2,1,4,3] => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 0
[2,3,1,4] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[2,3,4,1] => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[2,4,1,3] => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[2,4,3,1] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[3,1,2,4] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[3,1,4,2] => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[3,2,1,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[3,2,4,1] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[3,4,1,2] => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 0
[3,4,2,1] => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[4,1,2,3] => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[4,1,3,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[4,2,1,3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[4,2,3,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[4,3,1,2] => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[4,3,2,1] => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 0
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,2,3,5,4] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[1,2,4,3,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[1,2,4,5,3] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,2,5,3,4] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,2,5,4,3] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[1,3,2,4,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[1,3,2,5,4] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[1,3,4,2,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,3,4,5,2] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,3,5,2,4] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,3,5,4,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,4,2,3,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,4,2,5,3] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,4,3,2,5] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[1,4,3,5,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,4,5,2,3] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[1,4,5,3,2] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
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.
The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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.