Identifier
-
Mp00230:
Integer partitions
—parallelogram polyomino⟶
Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000485: Permutations ⟶ ℤ
Values
[2] => [1,0,1,0] => [1,2] => 1
[1,1] => [1,1,0,0] => [2,1] => 2
[3] => [1,0,1,0,1,0] => [1,2,3] => 1
[2,1] => [1,0,1,1,0,0] => [1,3,2] => 2
[1,1,1] => [1,1,0,1,0,0] => [2,3,1] => 3
[4] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 1
[3,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 2
[2,2] => [1,1,1,0,0,0] => [3,2,1] => 2
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 3
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => 4
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 1
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 2
[3,2] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 2
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 3
[2,2,1] => [1,1,1,0,0,1,0,0] => [3,2,4,1] => 3
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 4
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 5
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 1
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 2
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 2
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 3
[3,3] => [1,1,1,0,1,0,0,0] => [3,4,2,1] => 4
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => 3
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 4
[2,2,2] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => 2
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => 4
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 5
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => 6
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7] => 1
[6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,7,6] => 2
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => 2
[5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,4,6,7,5] => 3
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => 4
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => 3
[4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,3,5,6,7,4] => 4
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => 5
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 2
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => 4
[3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,4,5,6,7,3] => 5
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 3
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [3,2,4,5,6,1] => 5
[2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,7,2] => 6
[6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,4,7,6,5] => 2
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => 4
[5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,3,6,5,7,4] => 3
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => 3
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => 5
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 2
[4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [1,2,5,4,6,7,3] => 4
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => 5
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [3,4,2,5,6,1] => 6
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => 3
[3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,3,5,6,7,2] => 5
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => 4
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [4,3,2,5,6,1] => 4
[6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,3,6,7,5,4] => 4
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => 3
[5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [1,2,5,6,4,7,3] => 5
[5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,3,7,6,5,4] => 2
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [3,4,5,2,6,1] => 4
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => 5
[4,3,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [1,4,5,3,6,7,2] => 6
[4,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [1,2,6,5,4,7,3] => 3
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 2
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [3,5,4,2,6,1] => 6
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,5,6,4,3,2] => 4
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [4,5,3,2,6,1] => 5
[6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [1,2,5,6,7,4,3] => 3
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,2,1] => 6
[5,4,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [1,4,5,6,3,7,2] => 4
[5,3,2] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [1,2,5,7,6,4,3] => 5
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,6,5,2,1] => 3
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => 2
[4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [1,4,6,5,3,7,2] => 6
[4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [1,2,6,7,5,4,3] => 4
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [5,4,3,2,6,1] => 3
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [3,5,6,4,2,1] => 3
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [4,5,6,3,2,1] => 4
[6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [1,4,5,6,7,3,2] => 6
[5,4,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [1,4,5,7,6,3,2] => 3
[5,3,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,2,7,6,5,4,3] => 2
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [3,6,5,4,2,1] => 5
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [5,4,6,3,2,1] => 6
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [5,6,4,3,2,1] => 4
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => 2
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Description
The length of the longest cycle of a permutation.
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
to 312-avoiding permutation
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