Identifier
-
Mp00230:
Integer partitions
—parallelogram polyomino⟶
Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤ
Values
[2] => [1,0,1,0] => [1,0,1,0] => [[1,0],[0,1]] => 2
[1,1] => [1,1,0,0] => [1,1,0,0] => [[0,1],[1,0]] => 0
[3] => [1,0,1,0,1,0] => [1,0,1,0,1,0] => [[1,0,0],[0,1,0],[0,0,1]] => 3
[2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => [[0,1,0],[1,0,0],[0,0,1]] => 1
[1,1,1] => [1,1,0,1,0,0] => [1,1,0,1,0,0] => [[0,1,0],[1,-1,1],[0,1,0]] => -1
[4] => [1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]] => 4
[3,1] => [1,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,0] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]] => 2
[2,2] => [1,1,1,0,0,0] => [1,1,1,0,0,0] => [[0,0,1],[0,1,0],[1,0,0]] => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0] => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]] => 0
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0] => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]] => -2
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 5
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 3
[3,2] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]] => 2
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 1
[2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0] => [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]] => 0
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => -1
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]] => -3
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 6
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 4
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 3
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 2
[3,3] => [1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]] => 2
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 1
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 0
[2,2,2] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]] => 0
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]] => -1
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]] => -2
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]] => -4
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 4
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0,1,0] => [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 3
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 2
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]] => 1
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]] => 1
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]] => 0
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => [[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]] => -1
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0,1,0,1,0] => [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 4
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]] => 3
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]] => 2
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 2
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,-1,1,0],[0,0,1,0,0]] => 1
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]] => 0
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]] => -1
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0,1,0] => [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]] => 4
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0,1,0] => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]] => 2
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]] => 1
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0,1,0] => [[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]] => 0
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]] => 2
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Description
The trace of an alternating sign matrix.
Map
to symmetric ASM
Description
The diagonally symmetric alternating sign matrix corresponding to 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.
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
reverse
Description
The reversal of a Dyck path.
This is the Dyck path obtained by reading the path backwards.
This is the Dyck path obtained by reading the path backwards.
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