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Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St000005
Mapping
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000005: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => [1,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => [1,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1]
 => [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]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => [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]
 => 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
[1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => [1]
 => [1,0]
 => 0
[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]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[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]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [[4,3],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => [1]
 => [1,0]
 => 0
[1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3],[1]]
 => [1]
 => [1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1],[1]]
 => [1]
 => [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]
 => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1,1],[1]]
 => [1]
 => [1,0]
 => 0
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1,1],[1]]
 => [1]
 => [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]
 => 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
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [[4,2,1],[1]]
 => [1]
 => [1,0]
 => 0
[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]
 => 0
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [[4,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1
[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]
 => 1
[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
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [[4,3,1],[1]]
 => [1]
 => [1,0]
 => 0
Description
The bounce statistic of a Dyck path. The '''bounce path''' D′ of a Dyck path D is the Dyck path obtained from D by starting at the end point (2n,0), traveling north-west until hitting D, then bouncing back south-west to the x-axis, and repeating this procedure until finally reaching the point (0,0). The points where D′ touches the x-axis are called '''bounce points''', and a bounce path is uniquely determined by its bounce points. This statistic is given by the sum of all i for which the bounce path D′ of D touches the x-axis at (2i,0). In particular, the bounce statistics of D and D′ coincide.