Identifier
-
Mp00180:
Integer compositions
—to ribbon⟶
Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤ
Values
[2,1] => [[2,2],[1]] => [1] => [1,0] => 0
[1,2,1] => [[2,2,1],[1]] => [1] => [1,0] => 0
[2,1,1] => [[2,2,2],[1,1]] => [1,1] => [1,1,0,0] => 1
[2,2] => [[3,2],[1]] => [1] => [1,0] => 0
[3,1] => [[3,3],[2]] => [2] => [1,0,1,0] => 0
[1,1,2,1] => [[2,2,1,1],[1]] => [1] => [1,0] => 0
[1,2,1,1] => [[2,2,2,1],[1,1]] => [1,1] => [1,1,0,0] => 1
[1,2,2] => [[3,2,1],[1]] => [1] => [1,0] => 0
[1,3,1] => [[3,3,1],[2]] => [2] => [1,0,1,0] => 0
[2,1,1,1] => [[2,2,2,2],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 2
[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] => 1
[2,3] => [[4,2],[1]] => [1] => [1,0] => 0
[3,1,1] => [[3,3,3],[2,2]] => [2,2] => [1,1,1,0,0,0] => 1
[3,2] => [[4,3],[2]] => [2] => [1,0,1,0] => 0
[4,1] => [[4,4],[3]] => [3] => [1,0,1,0,1,0] => 0
[1,1,1,2,1] => [[2,2,1,1,1],[1]] => [1] => [1,0] => 0
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [1,1] => [1,1,0,0] => 1
[1,1,2,2] => [[3,2,1,1],[1]] => [1] => [1,0] => 0
[1,1,3,1] => [[3,3,1,1],[2]] => [2] => [1,0,1,0] => 0
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 2
[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] => 1
[1,2,3] => [[4,2,1],[1]] => [1] => [1,0] => 0
[1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => [1,1,1,0,0,0] => 1
[1,3,2] => [[4,3,1],[2]] => [2] => [1,0,1,0] => 0
[1,4,1] => [[4,4,1],[3]] => [3] => [1,0,1,0,1,0] => 0
[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] => 3
[2,1,1,2] => [[3,2,2,2],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 2
[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] => 1
[2,3,1] => [[4,4,2],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 1
[2,4] => [[5,2],[1]] => [1] => [1,0] => 0
[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] => 1
[3,2,1] => [[4,4,3],[3,2]] => [3,2] => [1,0,1,1,1,0,0,0] => 1
[3,3] => [[5,3],[2]] => [2] => [1,0,1,0] => 0
[4,1,1] => [[4,4,4],[3,3]] => [3,3] => [1,1,1,0,1,0,0,0] => 1
[4,2] => [[5,4],[3]] => [3] => [1,0,1,0,1,0] => 0
[5,1] => [[5,5],[4]] => [4] => [1,0,1,0,1,0,1,0] => 0
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => [1] => [1,0] => 0
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => [1,1] => [1,1,0,0] => 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]] => [1] => [1,0] => 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]] => [2] => [1,0,1,0] => 0
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 2
[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] => 1
[1,1,2,3] => [[4,2,1,1],[1]] => [1] => [1,0] => 0
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [2,2] => [1,1,1,0,0,0] => 1
[1,1,3,2] => [[4,3,1,1],[2]] => [2] => [1,0,1,0] => 0
[1,1,4,1] => [[4,4,1,1],[3]] => [3] => [1,0,1,0,1,0] => 0
[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] => 3
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 2
[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] => 1
[1,2,3,1] => [[4,4,2,1],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 1
[1,2,4] => [[5,2,1],[1]] => [1] => [1,0] => 0
[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] => 1
[1,3,2,1] => [[4,4,3,1],[3,2]] => [3,2] => [1,0,1,1,1,0,0,0] => 1
[1,3,3] => [[5,3,1],[2]] => [2] => [1,0,1,0] => 0
[1,4,1,1] => [[4,4,4,1],[3,3]] => [3,3] => [1,1,1,0,1,0,0,0] => 1
[1,4,2] => [[5,4,1],[3]] => [3] => [1,0,1,0,1,0] => 0
[1,5,1] => [[5,5,1],[4]] => [4] => [1,0,1,0,1,0,1,0] => 0
[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] => 4
[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] => 3
[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] => 3
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 2
[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] => 3
[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] => 2
[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] => 3
[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] => 2
[2,2,3] => [[5,3,2],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1
[2,3,1,1] => [[4,4,4,2],[3,3,1]] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 2
[2,3,2] => [[5,4,2],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 1
[2,4,1] => [[5,5,2],[4,1]] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[2,5] => [[6,2],[1]] => [1] => [1,0] => 0
[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] => 3
[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] => 2
[3,1,3] => [[5,3,3],[2,2]] => [2,2] => [1,1,1,0,0,0] => 1
[3,2,1,1] => [[4,4,4,3],[3,3,2]] => [3,3,2] => [1,1,1,0,1,1,0,0,0,0] => 2
[3,2,2] => [[5,4,3],[3,2]] => [3,2] => [1,0,1,1,1,0,0,0] => 1
[3,3,1] => [[5,5,3],[4,2]] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[3,4] => [[6,3],[2]] => [2] => [1,0,1,0] => 0
[4,1,1,1] => [[4,4,4,4],[3,3,3]] => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => 2
[4,1,2] => [[5,4,4],[3,3]] => [3,3] => [1,1,1,0,1,0,0,0] => 1
[4,2,1] => [[5,5,4],[4,3]] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 1
[4,3] => [[6,4],[3]] => [3] => [1,0,1,0,1,0] => 0
[5,1,1] => [[5,5,5],[4,4]] => [4,4] => [1,1,1,0,1,0,1,0,0,0] => 1
[5,2] => [[6,5],[4]] => [4] => [1,0,1,0,1,0,1,0] => 0
[6,1] => [[6,6],[5]] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[1,1,1,1,1,2,1] => [[2,2,1,1,1,1,1],[1]] => [1] => [1,0] => 0
[1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]] => [1,1] => [1,1,0,0] => 1
>>> Load all 186 entries. <<<
search for individual values
searching the database for the individual values of this statistic
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
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.
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.
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.
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.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!