Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St001032
Mapping
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001032: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3,1],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [[4,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [[4,4,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [[4,4,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [[4,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [[3,3,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [[4,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [[4,4,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [[4,4,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [[3,3,3,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [[4,3,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [[3,3,3,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [[4,4,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,1,0,0]
 => [[4,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => [[4,4,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [[5,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[1,1,0,1,0,0,1,1,1,0,0,0]
 => [[4,4,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 1
Description
The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. In other words, this is the number of valleys and peaks whose first step is in odd position, the initial position equal to 1. The generating function is given in [1].