- FindStat

Identifier

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,4] => [[6,3],[2]] => [2] => [1,0,1,0] => 2

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

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

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

[4,3] => [[6,4],[3]] => [3] => [1,0,1,0,1,0] => 3

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

[5,2] => [[6,5],[4]] => [4] => [1,0,1,0,1,0,1,0] => 4

[6,1] => [[6,6],[5]] => [5] => [1,0,1,0,1,0,1,0,1,0] => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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search for individual values

searching the database for the individual values of this statistic

Description

The maximal height of a column in the parallelogram polyomino associated with a Dyck path.

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.

Map

to ribbon

Description

The ribbon shape corresponding to an integer composition.
For an integer composition

$(a_1, \dots, a_n)$ , this is the ribbon shape whose

$i$ th row from the bottom has

$a_i$ cells.