Your data matches 29 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000065
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000065: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1,0]
 => [[1]]
 => 0
[2]
 => [1,0,1,0]
 => [1,1,0,0]
 => [[0,1],[1,0]]
 => 0
[1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => [[1,0],[0,1]]
 => 0
[3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [[0,0,1],[0,1,0],[1,0,0]]
 => 0
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => 0
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => 2
[2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
 => 0
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => 3
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => 2
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,-1,1,0],[0,0,1,0,0]]
 => 2
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
 => 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => 0
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => 0
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
 => 4
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => 4
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[1,0,-1,1,0,0],[0,0,1,0,0,0]]
 => 3
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => 0
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => 3
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[1,0,0,-1,1,0],[0,0,0,1,0,0]]
 => 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 0
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,-1,1],[0,0,0,0,1,0]]
 => 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
 => 0
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
 => 0
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,-1,1],[1,-1,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0]]
 => 6
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => 2
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[1,-1,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
 => 5
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 3
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [[0,0,1,0,0,0],[0,1,-1,0,1,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]]
 => 3
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 2
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
 => 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [[0,0,1,0,0,0],[0,1,-1,0,1,0],[1,-1,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
 => 4
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => 0
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[1,0,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
 => 3
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,-1,1,0],[0,1,-1,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
 => 6
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 0
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
 => 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
 => 5
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 0
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
 => 2
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[0,0,0,0,1,0]]
 => 2
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 3
Description
The number of entries equal to -1 in an alternating sign matrix. The number of nonzero entries, [[St000890]] is twice this number plus the dimension of the matrix.
Matching statistic: St000355
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1,0]
 => [1] => 0
[2]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,2] => 0
[1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => [2,1] => 0
[3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => 0
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,1,3] => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [3,1,2] => 0
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 2
[2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [3,2,1] => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => 3
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 2
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => 2
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 0
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => 0
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => 4
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => 4
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [3,1,2,4,5,6] => 3
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 0
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => 3
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [4,1,2,3,5,6] => 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 0
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,1,2,3,4,6] => 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => 0
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7] => 0
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [3,2,1,4,5,6] => 6
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => 2
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [4,2,1,3,5,6] => 5
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => 3
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [5,2,3,1,4,6] => 3
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => 2
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [4,2,3,1,5,6] => 4
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 0
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [5,3,1,2,4,6] => 3
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [4,3,2,1,5,6] => 6
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => 0
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [5,3,2,1,4,6] => 5
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => 0
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [6,3,4,2,1,5] => 2
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [5,2,3,4,1,6] => 2
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [5,4,2,3,1,6] => 3
Description
The number of occurrences of the pattern 21-3. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.
Matching statistic: St000359
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1,0]
 => [1] => 0
[2]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2,1] => 0
[1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,2] => 0
[3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => 0
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 0
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 2
[2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 0
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 3
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 2
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => 2
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 0
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 0
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [5,6,4,3,2,1] => 4
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => 4
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [5,4,6,3,2,1] => 3
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => 3
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [5,4,3,6,2,1] => 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,4,3,2,6,1] => 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => 0
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [7,6,5,4,3,2,1] => 0
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [4,5,6,3,2,1] => 6
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 2
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [4,5,3,6,2,1] => 5
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 3
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [3,5,4,2,6,1] => 3
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 2
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [2,5,4,3,1,6] => 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,5,4,6,2,1] => 4
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 0
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,3,5,2,6,1] => 3
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,2,1] => 6
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [3,2,5,4,1,6] => 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,4,5,2,6,1] => 5
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [2,3,5,4,1,6] => 2
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,5,4,3,6,1] => 2
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [4,3,2,5,1,6] => 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,3,5,6,1] => 3
Description
The number of occurrences of the pattern 23-1. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $23\!\!-\!\!1$.
Matching statistic: St000496
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000496: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => {{1}}
 => {{1}}
 => 0
[2]
 => [1,0,1,0]
 => {{1},{2}}
 => {{1,2}}
 => 0
[1,1]
 => [1,1,0,0]
 => {{1,2}}
 => {{1},{2}}
 => 0
[3]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => {{1,2,3}}
 => 0
[2,1]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => {{1,3},{2}}
 => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => {{1},{2,3}}
 => 0
[4]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => {{1,2,3,4}}
 => 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => {{1,3,4},{2}}
 => 2
[2,2]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => {{1},{2},{3}}
 => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => {{1,4},{2,3}}
 => 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => {{1},{2,3,4}}
 => 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => {{1,2,3,4,5}}
 => 0
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => {{1,3,4,5},{2}}
 => 3
[3,2]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => {{1,4},{2},{3}}
 => 2
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => {{1,4,5},{2,3}}
 => 2
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => {{1},{2,4},{3}}
 => 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => {{1,5},{2,3,4}}
 => 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => {{1},{2,3,4,5}}
 => 0
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5},{6}}
 => {{1,2,3,4,5,6}}
 => 0
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4},{5,6}}
 => {{1,3,4,5,6},{2}}
 => 4
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => {{1,4,5},{2},{3}}
 => 4
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3},{4,6},{5}}
 => {{1,4,5,6},{2,3}}
 => 3
[3,3]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => {{1},{2,3},{4}}
 => 0
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => {{1,5},{2,4},{3}}
 => 3
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => {{1},{2},{3,6},{4},{5}}
 => {{1,5,6},{2,3,4}}
 => 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => 0
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => {{1,5},{2,3},{4}}
 => {{1},{2,3,5},{4}}
 => 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => {{1},{2,6},{3},{4},{5}}
 => {{1,6},{2,3,4,5}}
 => 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,6},{2},{3},{4},{5}}
 => {{1},{2,3,4,5,6}}
 => 0
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5},{6},{7}}
 => {{1,2,3,4,5,6,7}}
 => 0
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3},{4,5,6}}
 => {{1,4,5,6},{2},{3}}
 => 6
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => {{1,5},{2,3},{4}}
 => 2
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => {{1},{2},{3,6},{4,5}}
 => {{1,5,6},{2,4},{3}}
 => 5
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => {{1,5},{2,4},{3}}
 => {{1},{2,5},{3,4}}
 => 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => {{1,5},{2},{3},{4}}
 => 3
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => {{1},{2,6},{3,4},{5}}
 => {{1,6},{2,3,5},{4}}
 => 3
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => {{1},{2,5},{3},{4}}
 => 2
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => {{1,6},{2,3},{4},{5}}
 => {{1},{2,3,4,6},{5}}
 => 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => {{1},{2},{3,4,6},{5}}
 => {{1,5,6},{2,3},{4}}
 => 4
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => {{1,2,5},{3},{4}}
 => {{1},{2,3,4},{5}}
 => 0
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => {{1},{2,6},{3,5},{4}}
 => {{1,6},{2,5},{3,4}}
 => 3
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => {{1},{2},{3,4,5,6}}
 => {{1,5,6},{2},{3},{4}}
 => 6
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => {{1,2,4,5},{3}}
 => {{1},{2},{3,4},{5}}
 => 0
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => {{1,6},{2,4},{3},{5}}
 => {{1},{2,3,6},{4,5}}
 => 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => {{1},{2,6},{3,4,5}}
 => {{1,6},{2,5},{3},{4}}
 => 5
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => {{1,2,3,5},{4}}
 => {{1},{2,3},{4},{5}}
 => 0
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => {{1,6},{2,3,4},{5}}
 => {{1},{2,3,6},{4},{5}}
 => 2
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => {{1},{2,3,6},{4},{5}}
 => {{1,6},{2,3,4},{5}}
 => 2
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => {{1,6},{2,5},{3},{4}}
 => {{1},{2,6},{3,4,5}}
 => 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => {{1},{2,3,5,6},{4}}
 => {{1,6},{2},{3,4},{5}}
 => 3
Description
The rcs statistic of a set partition. Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$. According to [1, Definition 3], a '''rcs''' (right-closer-smaller) of $S$ is given by a pair $i > j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a < b$.
Matching statistic: St001033
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1,0]
 => [1,0]
 => 0
[2]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 0
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 3
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 6
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 5
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 3
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 3
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 4
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 0
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 3
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 0
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 5
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 0
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 3
Description
The normalized area of the parallelogram polyomino associated with the Dyck path. The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path. The area itself is equidistributed with [[St001034]] and with [[St000395]].
Matching statistic: St000609
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000609: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => {{1}}
 => ? = 0
[2]
 => [1,0,1,0]
 => {{1},{2}}
 => 0
[1,1]
 => [1,1,0,0]
 => {{1,2}}
 => 0
[3]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0
[2,1]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 0
[4]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 2
[2,2]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 0
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 3
[3,2]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 2
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => 2
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => 0
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5},{6}}
 => 0
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4},{5,6}}
 => 4
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 4
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3},{4,6},{5}}
 => 3
[3,3]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 0
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => 3
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => {{1},{2},{3,6},{4},{5}}
 => 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 0
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => {{1,5},{2,3},{4}}
 => 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => {{1},{2,6},{3},{4},{5}}
 => 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,6},{2},{3},{4},{5}}
 => 0
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5},{6},{7}}
 => 0
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3},{4,5,6}}
 => 6
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => 2
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => {{1},{2},{3,6},{4,5}}
 => 5
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => {{1,5},{2,4},{3}}
 => 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 3
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => {{1},{2,6},{3,4},{5}}
 => 3
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => 2
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => {{1,6},{2,3},{4},{5}}
 => 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => {{1},{2},{3,4,6},{5}}
 => 4
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => {{1,2,5},{3},{4}}
 => 0
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => {{1},{2,6},{3,5},{4}}
 => 3
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => {{1},{2},{3,4,5,6}}
 => 6
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => {{1,2,4,5},{3}}
 => 0
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => {{1,6},{2,4},{3},{5}}
 => 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => {{1},{2,6},{3,4,5}}
 => 5
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => {{1,2,3,5},{4}}
 => 0
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => {{1,6},{2,3,4},{5}}
 => 2
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => {{1},{2,3,6},{4},{5}}
 => 2
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => {{1,6},{2,5},{3},{4}}
 => 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => {{1},{2,3,5,6},{4}}
 => 3
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 0
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.
Matching statistic: St000039
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1,0]
 => [1] => 0
[2]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2,1] => 0
[1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,2] => 0
[3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3,1,2] => 0
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 0
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 2
[2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => 0
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => 3
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 2
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => 2
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => 0
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,1,2,3,4,5] => 0
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [5,6,1,2,3,4] => 4
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => 4
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [5,1,6,2,3,4] => 3
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => 3
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [5,1,2,6,3,4] => 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,1,2,3,6,4] => 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,1,2,3,4,6] => 0
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [7,1,2,3,4,5,6] => ? = 0
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [4,5,6,1,2,3] => 6
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => 2
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [4,5,1,6,2,3] => 5
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 3
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [3,5,1,2,6,4] => 3
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 2
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [2,5,1,3,4,6] => 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,5,1,6,2,4] => 4
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => 0
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,1,5,2,6,3] => 3
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,1,2] => 6
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [3,1,5,2,4,6] => 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,4,5,1,6,2] => 5
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [2,3,5,1,4,6] => 2
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,5,1,3,6,4] => 2
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [4,1,2,5,3,6] => 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,1,5,6,3] => 3
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0
Description
The number of crossings of a permutation. A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$. Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
Matching statistic: St000491
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000491: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => {{1}}
 => {{1}}
 => ? = 0
[2]
 => [1,0,1,0]
 => {{1},{2}}
 => {{1,2}}
 => 0
[1,1]
 => [1,1,0,0]
 => {{1,2}}
 => {{1},{2}}
 => 0
[3]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => {{1,2,3}}
 => 0
[2,1]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => {{1,3},{2}}
 => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => {{1},{2,3}}
 => 0
[4]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => {{1,2,3,4}}
 => 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => {{1,3,4},{2}}
 => 2
[2,2]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => {{1},{2},{3}}
 => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => {{1,4},{2,3}}
 => 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => {{1},{2,3,4}}
 => 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => {{1,2,3,4,5}}
 => 0
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => {{1,3,4,5},{2}}
 => 3
[3,2]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => {{1,4},{2},{3}}
 => 2
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => {{1,4,5},{2,3}}
 => 2
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => {{1},{2,4},{3}}
 => 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => {{1,5},{2,3,4}}
 => 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => {{1},{2,3,4,5}}
 => 0
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5},{6}}
 => {{1,2,3,4,5,6}}
 => 0
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4},{5,6}}
 => {{1,3,4,5,6},{2}}
 => 4
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => {{1,4,5},{2},{3}}
 => 4
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3},{4,6},{5}}
 => {{1,4,5,6},{2,3}}
 => 3
[3,3]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => {{1},{2,3},{4}}
 => 0
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => {{1,5},{2,4},{3}}
 => 3
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => {{1},{2},{3,6},{4},{5}}
 => {{1,5,6},{2,3,4}}
 => 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => 0
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => {{1,5},{2,3},{4}}
 => {{1},{2,3,5},{4}}
 => 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => {{1},{2,6},{3},{4},{5}}
 => {{1,6},{2,3,4,5}}
 => 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,6},{2},{3},{4},{5}}
 => {{1},{2,3,4,5,6}}
 => 0
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5},{6},{7}}
 => {{1,2,3,4,5,6,7}}
 => 0
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3},{4,5,6}}
 => {{1,4,5,6},{2},{3}}
 => 6
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => {{1,5},{2,3},{4}}
 => 2
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => {{1},{2},{3,6},{4,5}}
 => {{1,5,6},{2,4},{3}}
 => 5
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => {{1,5},{2,4},{3}}
 => {{1},{2,5},{3,4}}
 => 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => {{1,5},{2},{3},{4}}
 => 3
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => {{1},{2,6},{3,4},{5}}
 => {{1,6},{2,3,5},{4}}
 => 3
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => {{1},{2,5},{3},{4}}
 => 2
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => {{1,6},{2,3},{4},{5}}
 => {{1},{2,3,4,6},{5}}
 => 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => {{1},{2},{3,4,6},{5}}
 => {{1,5,6},{2,3},{4}}
 => 4
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => {{1,2,5},{3},{4}}
 => {{1},{2,3,4},{5}}
 => 0
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => {{1},{2,6},{3,5},{4}}
 => {{1,6},{2,5},{3,4}}
 => 3
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => {{1},{2},{3,4,5,6}}
 => {{1,5,6},{2},{3},{4}}
 => 6
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => {{1,2,4,5},{3}}
 => {{1},{2},{3,4},{5}}
 => 0
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => {{1,6},{2,4},{3},{5}}
 => {{1},{2,3,6},{4,5}}
 => 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => {{1},{2,6},{3,4,5}}
 => {{1,6},{2,5},{3},{4}}
 => 5
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => {{1,2,3,5},{4}}
 => {{1},{2,3},{4},{5}}
 => 0
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => {{1,6},{2,3,4},{5}}
 => {{1},{2,3,6},{4},{5}}
 => 2
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => {{1},{2,3,6},{4},{5}}
 => {{1,6},{2,3,4},{5}}
 => 2
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => {{1,6},{2,5},{3},{4}}
 => {{1},{2,6},{3,4,5}}
 => 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => {{1},{2,3,5,6},{4}}
 => {{1,6},{2},{3,4},{5}}
 => 3
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => {{1},{2},{3},{4},{5}}
 => 0
Description
The number of inversions of a set partition. Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$. According to [1], see also [2,3], an inversion of $S$ is given by a pair $i > j$ such that $j = \operatorname{min} B_b$ and $i \in B_a$ for $a < b$. This statistic is called '''ros''' in [1, Definition 3] for "right, opener, smaller". This is also the number of occurrences of the pattern {{1, 3}, {2}} such that 1 and 2 are minimal elements of blocks.
Matching statistic: St000572
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
St000572: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => {{1}}
 => {{1}}
 => ? = 0
[2]
 => [1,0,1,0]
 => {{1},{2}}
 => {{1},{2}}
 => 0
[1,1]
 => [1,1,0,0]
 => {{1,2}}
 => {{1,2}}
 => 0
[3]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
[2,1]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => {{1,3},{2}}
 => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => {{1},{2,3}}
 => 0
[4]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 2
[2,2]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => {{1,2,3}}
 => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => {{1,5},{2},{3},{4}}
 => 3
[3,2]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => {{1,3},{2,4}}
 => 2
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => {{1},{2,5},{3},{4}}
 => 2
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => {{1,3,4},{2}}
 => 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => {{1},{2},{3,5},{4}}
 => 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => {{1},{2},{3},{4,5}}
 => 0
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5},{6}}
 => {{1},{2},{3},{4},{5},{6}}
 => 0
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4},{5,6}}
 => {{1,6},{2},{3},{4},{5}}
 => 4
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => {{1,4},{2,5},{3}}
 => 4
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3},{4,6},{5}}
 => {{1},{2,6},{3},{4},{5}}
 => 3
[3,3]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => {{1,2},{3,4}}
 => 0
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => {{1,4},{2},{3,5}}
 => 3
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => {{1},{2},{3,6},{4},{5}}
 => {{1},{2},{3,6},{4},{5}}
 => 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => 0
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => {{1,5},{2,3},{4}}
 => {{1,3},{2},{4,5}}
 => 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => {{1},{2,6},{3},{4},{5}}
 => {{1},{2},{3},{4,6},{5}}
 => 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,6},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5,6}}
 => 0
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5},{6},{7}}
 => {{1},{2},{3},{4},{5},{6},{7}}
 => 0
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3},{4,5,6}}
 => {{1,5},{2,6},{3},{4}}
 => 6
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => {{1,3,5},{2},{4}}
 => 2
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => {{1},{2},{3,6},{4,5}}
 => {{1,5},{2},{3,6},{4}}
 => 5
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => {{1,5},{2,4},{3}}
 => {{1},{2,4,5},{3}}
 => 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => {{1,3,5},{2,4}}
 => 3
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => {{1},{2,6},{3,4},{5}}
 => {{1,4,6},{2},{3},{5}}
 => 3
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => {{1,3},{2,4,5}}
 => 2
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => {{1,6},{2,3},{4},{5}}
 => {{1,3},{2},{4},{5,6}}
 => 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => {{1},{2},{3,4,6},{5}}
 => {{1,4},{2},{3,6},{5}}
 => 4
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => {{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => 0
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => {{1},{2,6},{3,5},{4}}
 => {{1},{2,5},{3},{4,6}}
 => 3
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => {{1},{2},{3,4,5,6}}
 => {{1,4},{2,5},{3,6}}
 => 6
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => {{1,2,4,5},{3}}
 => {{1,2},{3,4,5}}
 => 0
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => {{1,6},{2,4},{3},{5}}
 => {{1},{2,4},{3},{5,6}}
 => 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => {{1},{2,6},{3,4,5}}
 => {{1,4,6},{2,5},{3}}
 => 5
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => {{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => 0
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => {{1,6},{2,3,4},{5}}
 => {{1,3},{2,4},{5,6}}
 => 2
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => {{1},{2,3,6},{4},{5}}
 => {{1,3},{2},{4,6},{5}}
 => 2
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => {{1,6},{2,5},{3},{4}}
 => {{1},{2},{3,5,6},{4}}
 => 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => {{1},{2,3,5,6},{4}}
 => {{1,3,5},{2},{4,6}}
 => 3
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => 0
Description
The dimension exponent of a set partition. This is $$\sum_{B\in\pi} (\max(B) - \min(B) + 1) - n$$ where the summation runs over the blocks of the set partition $\pi$ of $\{1,\dots,n\}$. It is thus equal to the difference [[St000728]] - [[St000211]]. This is also the number of occurrences of the pattern {{1, 3}, {2}}, such that 1 and 3 are consecutive elements in a block. This is also the number of occurrences of the pattern {{1, 3}, {2}}, such that 1 is the minimal and 3 is the maximal element of the block.
Matching statistic: St000581
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000581: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => {{1}}
 => {{1}}
 => ? = 0
[2]
 => [1,0,1,0]
 => {{1},{2}}
 => {{1,2}}
 => 0
[1,1]
 => [1,1,0,0]
 => {{1,2}}
 => {{1},{2}}
 => 0
[3]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => {{1,2,3}}
 => 0
[2,1]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => {{1,3},{2}}
 => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => {{1},{2,3}}
 => 0
[4]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => {{1,2,3,4}}
 => 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => {{1,3,4},{2}}
 => 2
[2,2]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => {{1},{2},{3}}
 => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => {{1,4},{2,3}}
 => 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => {{1},{2,3,4}}
 => 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => {{1,2,3,4,5}}
 => 0
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => {{1,3,4,5},{2}}
 => 3
[3,2]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => {{1,4},{2},{3}}
 => 2
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => {{1,4,5},{2,3}}
 => 2
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => {{1},{2,4},{3}}
 => 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => {{1,5},{2,3,4}}
 => 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => {{1},{2,3,4,5}}
 => 0
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5},{6}}
 => {{1,2,3,4,5,6}}
 => 0
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4},{5,6}}
 => {{1,3,4,5,6},{2}}
 => 4
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => {{1,4,5},{2},{3}}
 => 4
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3},{4,6},{5}}
 => {{1,4,5,6},{2,3}}
 => 3
[3,3]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => {{1},{2,3},{4}}
 => 0
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => {{1,5},{2,4},{3}}
 => 3
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => {{1},{2},{3,6},{4},{5}}
 => {{1,5,6},{2,3,4}}
 => 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => 0
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => {{1,5},{2,3},{4}}
 => {{1},{2,3,5},{4}}
 => 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => {{1},{2,6},{3},{4},{5}}
 => {{1,6},{2,3,4,5}}
 => 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,6},{2},{3},{4},{5}}
 => {{1},{2,3,4,5,6}}
 => 0
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5},{6},{7}}
 => {{1,2,3,4,5,6,7}}
 => 0
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3},{4,5,6}}
 => {{1,4,5,6},{2},{3}}
 => 6
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => {{1,5},{2,3},{4}}
 => 2
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => {{1},{2},{3,6},{4,5}}
 => {{1,5,6},{2,4},{3}}
 => 5
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => {{1,5},{2,4},{3}}
 => {{1},{2,5},{3,4}}
 => 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => {{1,5},{2},{3},{4}}
 => 3
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => {{1},{2,6},{3,4},{5}}
 => {{1,6},{2,3,5},{4}}
 => 3
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => {{1},{2,5},{3},{4}}
 => 2
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => {{1,6},{2,3},{4},{5}}
 => {{1},{2,3,4,6},{5}}
 => 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => {{1},{2},{3,4,6},{5}}
 => {{1,5,6},{2,3},{4}}
 => 4
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => {{1,2,5},{3},{4}}
 => {{1},{2,3,4},{5}}
 => 0
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => {{1},{2,6},{3,5},{4}}
 => {{1,6},{2,5},{3,4}}
 => 3
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => {{1},{2},{3,4,5,6}}
 => {{1,5,6},{2},{3},{4}}
 => 6
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => {{1,2,4,5},{3}}
 => {{1},{2},{3,4},{5}}
 => 0
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => {{1,6},{2,4},{3},{5}}
 => {{1},{2,3,6},{4,5}}
 => 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => {{1},{2,6},{3,4,5}}
 => {{1,6},{2,5},{3},{4}}
 => 5
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => {{1,2,3,5},{4}}
 => {{1},{2,3},{4},{5}}
 => 0
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => {{1,6},{2,3,4},{5}}
 => {{1},{2,3,6},{4},{5}}
 => 2
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => {{1},{2,3,6},{4},{5}}
 => {{1,6},{2,3,4},{5}}
 => 2
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => {{1,6},{2,5},{3},{4}}
 => {{1},{2,6},{3,4,5}}
 => 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => {{1},{2,3,5,6},{4}}
 => {{1,6},{2},{3,4},{5}}
 => 3
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => {{1},{2},{3},{4},{5}}
 => 0
Description
The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal.
The following 19 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001811The Castelnuovo-Mumford regularity of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001822The number of alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000068The number of minimal elements in a poset. St000090The variation of a composition. St000091The descent variation of a composition. St000233The number of nestings of a set partition. St000650The number of 3-rises of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001722The number of minimal chains with small intervals between a binary word and the top element. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001488The number of corners of a skew partition.