Identifier
-
Mp00180:
Integer compositions
—to ribbon⟶
Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000120: Dyck paths ⟶ ℤ
Values
[1] => [[1],[]] => [1] => [1,0,1,0] => 1
[1,1] => [[1,1],[]] => [1,1] => [1,0,1,1,0,0] => 1
[2] => [[2],[]] => [2] => [1,1,0,0,1,0] => 2
[1,1,1] => [[1,1,1],[]] => [1,1,1] => [1,0,1,1,1,0,0,0] => 1
[1,2] => [[2,1],[]] => [2,1] => [1,0,1,0,1,0] => 1
[2,1] => [[2,2],[1]] => [2,2] => [1,1,0,0,1,1,0,0] => 2
[3] => [[3],[]] => [3] => [1,1,1,0,0,0,1,0] => 3
[1,1,1,1] => [[1,1,1,1],[]] => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => 1
[1,1,2] => [[2,1,1],[]] => [2,1,1] => [1,0,1,1,0,1,0,0] => 1
[1,2,1] => [[2,2,1],[1]] => [2,2,1] => [1,0,1,0,1,1,0,0] => 2
[1,3] => [[3,1],[]] => [3,1] => [1,1,0,1,0,0,1,0] => 2
[2,1,1] => [[2,2,2],[1,1]] => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => 2
[2,2] => [[3,2],[1]] => [3,2] => [1,1,0,0,1,0,1,0] => 2
[3,1] => [[3,3],[2]] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 3
[4] => [[4],[]] => [4] => [1,1,1,1,0,0,0,0,1,0] => 4
[1,1,1,1,1] => [[1,1,1,1,1],[]] => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
[1,1,1,2] => [[2,1,1,1],[]] => [2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => 1
[1,1,2,1] => [[2,2,1,1],[1]] => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => 2
[1,1,3] => [[3,1,1],[]] => [3,1,1] => [1,0,1,1,0,0,1,0] => 1
[1,2,1,1] => [[2,2,2,1],[1,1]] => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => 2
[1,2,2] => [[3,2,1],[1]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 2
[1,3,1] => [[3,3,1],[2]] => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => 3
[1,4] => [[4,1],[]] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => 3
[2,1,1,1] => [[2,2,2,2],[1,1,1]] => [2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2
[2,1,2] => [[3,2,2],[1,1]] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => 2
[2,2,1] => [[3,3,2],[2,1]] => [3,3,2] => [1,1,0,0,1,0,1,1,0,0] => 2
[2,3] => [[4,2],[1]] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 3
[3,1,1] => [[3,3,3],[2,2]] => [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
[3,2] => [[4,3],[2]] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 3
[4,1] => [[4,4],[3]] => [4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4
[5] => [[5],[]] => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
[1,1,1,1,2] => [[2,1,1,1,1],[]] => [2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]] => [2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2
[1,1,1,3] => [[3,1,1,1],[]] => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0] => 2
[1,1,2,2] => [[3,2,1,1],[1]] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => 2
[1,1,3,1] => [[3,3,1,1],[2]] => [3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => 3
[1,1,4] => [[4,1,1],[]] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 2
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0] => 2
[1,2,1,2] => [[3,2,2,1],[1,1]] => [3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => 2
[1,2,2,1] => [[3,3,2,1],[2,1]] => [3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => 2
[1,2,3] => [[4,2,1],[1]] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => 3
[1,3,1,1] => [[3,3,3,1],[2,2]] => [3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0] => 3
[1,3,2] => [[4,3,1],[2]] => [4,3,1] => [1,1,0,1,0,0,1,0,1,0] => 3
[1,4,1] => [[4,4,1],[3]] => [4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0] => 4
[1,5] => [[5,1],[]] => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => 4
[2,1,1,2] => [[3,2,2,2],[1,1,1]] => [3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,0] => 2
[2,1,2,1] => [[3,3,2,2],[2,1,1]] => [3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => 2
[2,1,3] => [[4,2,2],[1,1]] => [4,2,2] => [1,1,0,0,1,1,0,0,1,0] => 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]] => [3,3,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0] => 3
[2,2,2] => [[4,3,2],[2,1]] => [4,3,2] => [1,1,0,0,1,0,1,0,1,0] => 2
[2,3,1] => [[4,4,2],[3,1]] => [4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,0] => 3
[2,4] => [[5,2],[1]] => [5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => 4
[3,1,2] => [[4,3,3],[2,2]] => [4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,0] => 3
[3,2,1] => [[4,4,3],[3,2]] => [4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0] => 3
[3,3] => [[5,3],[2]] => [5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => 4
[4,2] => [[5,4],[3]] => [5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 4
[1,1,1,1,3] => [[3,1,1,1,1],[]] => [3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]] => [3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => 2
[1,1,1,3,1] => [[3,3,1,1,1],[2]] => [3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => 3
[1,1,1,4] => [[4,1,1,1],[]] => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => [3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => 2
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,0] => 2
[1,1,2,3] => [[4,2,1,1],[1]] => [4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => 2
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0] => 3
[1,1,3,2] => [[4,3,1,1],[2]] => [4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => 3
[1,1,4,1] => [[4,4,1,1],[3]] => [4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,0] => 4
[1,1,5] => [[5,1,1],[]] => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 3
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => [3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,0] => 2
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => [3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,0,0] => 2
[1,2,1,3] => [[4,2,2,1],[1,1]] => [4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => [3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,0] => 3
[1,2,2,2] => [[4,3,2,1],[2,1]] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 2
[1,2,3,1] => [[4,4,2,1],[3,1]] => [4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,0] => 3
[1,2,4] => [[5,2,1],[1]] => [5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => 4
[1,3,1,2] => [[4,3,3,1],[2,2]] => [4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,0] => 3
[1,3,2,1] => [[4,4,3,1],[3,2]] => [4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,0,0] => 3
[1,3,3] => [[5,3,1],[2]] => [5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => 4
[1,4,2] => [[5,4,1],[3]] => [5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0] => 4
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => [4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0] => 2
[2,1,2,2] => [[4,3,2,2],[2,1,1]] => [4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,0] => 2
[2,1,3,1] => [[4,4,2,2],[3,1,1]] => [4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 2
[2,1,4] => [[5,2,2],[1,1]] => [5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,0] => 3
[2,2,1,2] => [[4,3,3,2],[2,2,1]] => [4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,0,0] => 3
[2,2,2,1] => [[4,4,3,2],[3,2,1]] => [4,4,3,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 3
[2,2,3] => [[5,3,2],[2,1]] => [5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => 3
[2,3,2] => [[5,4,2],[3,1]] => [5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,0] => 3
[3,1,3] => [[5,3,3],[2,2]] => [5,3,3] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
[3,2,2] => [[5,4,3],[3,2]] => [5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,0] => 3
[1,1,1,1,4] => [[4,1,1,1,1],[]] => [4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => 1
[1,1,1,3,2] => [[4,3,1,1,1],[2]] => [4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 3
[1,1,1,5] => [[5,1,1,1],[]] => [5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,0] => 2
[1,1,2,2,2] => [[4,3,2,1,1],[2,1]] => [4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 2
[1,1,2,3,1] => [[4,4,2,1,1],[3,1]] => [4,4,2,1,1] => [1,0,1,1,0,1,0,0,1,1,0,0] => 3
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]] => [4,3,3,1,1] => [1,0,1,1,0,0,1,1,0,1,0,0] => 3
[1,1,3,2,1] => [[4,4,3,1,1],[3,2]] => [4,4,3,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0] => 3
[1,1,3,3] => [[5,3,1,1],[2]] => [5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => 4
[1,2,1,2,2] => [[4,3,2,2,1],[2,1,1]] => [4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
[1,2,1,3,1] => [[4,4,2,2,1],[3,1,1]] => [4,4,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0] => 2
[1,2,2,1,2] => [[4,3,3,2,1],[2,2,1]] => [4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 3
[1,2,2,2,1] => [[4,4,3,2,1],[3,2,1]] => [4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 3
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Description
The number of left tunnels of a Dyck path.
A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b<n then the tunnel is called left.
A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b<n then the tunnel is called left.
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
outer shape
Description
The outer shape of the skew partition.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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