Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St000946
Mapping
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000946: 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]
 => 1
[1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 0
[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]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 1
[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]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 0
[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]
 => 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 0
[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]
 => 1
[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]
 => 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 1
[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]
 => 0
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [[4,4,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [[4,4,1],[2]]
 => [2]
 => [1,0,1,0]
 => 0
[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]
 => 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 1
[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]
 => 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 0
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 0
[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]
 => 1
[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]
 => 1
[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]
 => 3
[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]
 => 2
[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]
 => 3
[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]
 => 2
[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]
 => 1
[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]
 => 1
[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]
 => 1
[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]
 => 0
[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 sum of the skew hook positions in a Dyck path. A skew hook is an occurrence of a down step followed by two up steps or of an up step followed by a down step. Write $U_i$ for the $i$-th up step and $D_j$ for the $j$-th down step in the Dyck path. Then the skew hook set is the set $$H = \{j: U_{i−1} U_i D_j \text{ is a skew hook}\} \cup \{i: D_{i−1} D_i U_j\text{ is a skew hook}\}.$$ This statistic is the sum of all elements in $H$.