- FindStat

Identifier

Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[5],[]] => [] => [] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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

parallelogram polyomino

Description

Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.

Map

inner shape

Description

The inner shape of a skew partition.