Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St000675
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,2],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[3,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[4,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[4,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[3,3,2],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[4,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3
[[5,2],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[4,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[5,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[5,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[[3,3,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[4,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[4,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[5,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[[3,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[4,3,2],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2
[[4,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[5,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[[3,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[3,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[4,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[[4,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[[4,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[5,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3
[[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[[5,4],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[4,4,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[[4,3,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[[5,4,1],[4,1]]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[4,4,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[[3,2,2],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[4,3,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
Description
The number of centered multitunnels of a Dyck path. This is the number of factorisations $D = A B C$ of a Dyck path, such that $B$ is a Dyck path and $A$ and $B$ have the same length.