- FindStat

Identifier

Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000676: Dyck paths ⟶ ℤ

Values

[[1],[]] => [1] => [1,0] => 1

[[2],[]] => [2] => [1,1,0,0] => 1

[[1,1],[]] => [1,1] => [1,0,1,0] => 2

[[2,1],[1]] => [1,1] => [1,0,1,0] => 2

[[3],[]] => [3] => [1,1,1,0,0,0] => 2

[[2,1],[]] => [2,1] => [1,1,0,0,1,0] => 2

[[3,1],[1]] => [2,1] => [1,1,0,0,1,0] => 2

[[2,2],[1]] => [1,2] => [1,0,1,1,0,0] => 2

[[3,2],[2]] => [1,2] => [1,0,1,1,0,0] => 2

[[1,1,1],[]] => [1,1,1] => [1,0,1,0,1,0] => 3

[[2,2,1],[1,1]] => [1,1,1] => [1,0,1,0,1,0] => 3

[[2,1,1],[1]] => [1,1,1] => [1,0,1,0,1,0] => 3

[[3,2,1],[2,1]] => [1,1,1] => [1,0,1,0,1,0] => 3

[[4],[]] => [4] => [1,1,1,1,0,0,0,0] => 2

[[3,1],[]] => [3,1] => [1,1,1,0,0,0,1,0] => 3

[[4,1],[1]] => [3,1] => [1,1,1,0,0,0,1,0] => 3

[[2,2],[]] => [2,2] => [1,1,0,0,1,1,0,0] => 2

[[3,2],[1]] => [2,2] => [1,1,0,0,1,1,0,0] => 2

[[4,2],[2]] => [2,2] => [1,1,0,0,1,1,0,0] => 2

[[2,1,1],[]] => [2,1,1] => [1,1,0,0,1,0,1,0] => 3

[[3,2,1],[1,1]] => [2,1,1] => [1,1,0,0,1,0,1,0] => 3

[[3,1,1],[1]] => [2,1,1] => [1,1,0,0,1,0,1,0] => 3

[[4,2,1],[2,1]] => [2,1,1] => [1,1,0,0,1,0,1,0] => 3

[[3,3],[2]] => [1,3] => [1,0,1,1,1,0,0,0] => 3

[[4,3],[3]] => [1,3] => [1,0,1,1,1,0,0,0] => 3

[[2,2,1],[1]] => [1,2,1] => [1,0,1,1,0,0,1,0] => 3

[[3,3,1],[2,1]] => [1,2,1] => [1,0,1,1,0,0,1,0] => 3

[[3,2,1],[2]] => [1,2,1] => [1,0,1,1,0,0,1,0] => 3

[[4,3,1],[3,1]] => [1,2,1] => [1,0,1,1,0,0,1,0] => 3

[[2,2,2],[1,1]] => [1,1,2] => [1,0,1,0,1,1,0,0] => 3

[[3,3,2],[2,2]] => [1,1,2] => [1,0,1,0,1,1,0,0] => 3

[[3,2,2],[2,1]] => [1,1,2] => [1,0,1,0,1,1,0,0] => 3

[[4,3,2],[3,2]] => [1,1,2] => [1,0,1,0,1,1,0,0] => 3

[[1,1,1,1],[]] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 4

[[2,2,2,1],[1,1,1]] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 4

[[2,2,1,1],[1,1]] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 4

[[3,3,2,1],[2,2,1]] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 4

[[2,1,1,1],[1]] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 4

[[3,2,2,1],[2,1,1]] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 4

[[3,2,1,1],[2,1]] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 4

[[4,3,2,1],[3,2,1]] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 4

[[5],[]] => [5] => [1,1,1,1,1,0,0,0,0,0] => 3

[[4,1],[]] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 3

[[5,1],[1]] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 3

[[3,2],[]] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 3

[[4,2],[1]] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 3

[[5,2],[2]] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 3

[[3,1,1],[]] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 4

[[4,2,1],[1,1]] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 4

[[4,1,1],[1]] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 4

[[5,2,1],[2,1]] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 4

[[3,3],[1]] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 3

[[4,3],[2]] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 3

[[5,3],[3]] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 3

[[2,2,1],[]] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3

[[3,3,1],[1,1]] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3

[[3,2,1],[1]] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3

[[4,3,1],[2,1]] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3

[[4,2,1],[2]] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3

[[5,3,1],[3,1]] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3

[[3,2,2],[1,1]] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 3

[[4,3,2],[2,2]] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 3

[[4,2,2],[2,1]] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 3

[[5,3,2],[3,2]] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 3

[[2,1,1,1],[]] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 4

[[3,2,2,1],[1,1,1]] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 4

[[3,2,1,1],[1,1]] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 4

[[4,3,2,1],[2,2,1]] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 4

[[3,1,1,1],[1]] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 4

[[4,2,2,1],[2,1,1]] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 4

[[4,2,1,1],[2,1]] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 4

[[5,3,2,1],[3,2,1]] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 4

[[4,4],[3]] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 3

[[5,4],[4]] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 3

[[3,3,1],[2]] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4

[[4,4,1],[3,1]] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4

[[4,3,1],[3]] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4

[[5,4,1],[4,1]] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4

[[2,2,2],[1]] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 3

[[3,3,2],[2,1]] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 3

[[4,4,2],[3,2]] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 3

[[3,2,2],[2]] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 3

[[4,3,2],[3,1]] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 3

[[5,4,2],[4,2]] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 3

[[2,2,1,1],[1]] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4

[[3,3,2,1],[2,1,1]] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4

[[3,3,1,1],[2,1]] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4

[[4,4,2,1],[3,2,1]] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4

[[3,2,1,1],[2]] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4

[[4,3,2,1],[3,1,1]] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4

[[4,3,1,1],[3,1]] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4

[[5,4,2,1],[4,2,1]] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4

[[3,3,3],[2,2]] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 4

[[4,4,3],[3,3]] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 4

[[4,3,3],[3,2]] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 4

[[5,4,3],[4,3]] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 4

[[2,2,2,1],[1,1]] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4

[[3,3,3,1],[2,2,1]] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4

[[3,3,2,1],[2,2]] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4

[[4,4,3,1],[3,3,1]] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4

[[3,2,2,1],[2,1]] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4

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Description

The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].

Map

row lengths

Description

The sequence of row lengths from top to bottom.

Map

bounce path

Description

The bounce path determined by an integer composition.