Identifier
Values
[1] => [1,0] => [2,1] => 2
[2] => [1,0,1,0] => [3,1,2] => 3
[1,1] => [1,1,0,0] => [2,3,1] => 3
[3] => [1,0,1,0,1,0] => [4,1,2,3] => 4
[2,1] => [1,0,1,1,0,0] => [3,1,4,2] => 5
[1,1,1] => [1,1,0,1,0,0] => [4,3,1,2] => 12
[4] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 5
[3,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 7
[2,2] => [1,1,1,0,0,0] => [2,3,4,1] => 4
[2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 15
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 20
[5] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 6
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 9
[3,2] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 7
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => 18
[2,2,1] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 18
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 24
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => 15
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => 7
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => 11
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 10
[3,3] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 30
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 30
[2,2,2] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 5
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 32
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => 42
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => 8
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => 13
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 36
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => 54
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 9
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 24
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [2,6,7,1,3,4,5] => 25
[1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [8,7,1,2,3,4,5,6] => 56
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => 120
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => 42
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => 60
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 6
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [7,6,4,5,1,2,3] => 420
[6,6] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [6,7,8,1,2,3,4,5] => 56
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => 7
[5,5,5] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [8,7,4,5,6,1,2,3] => 1120
[4,4,4,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => 8
[] => [] => [1] => 1
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Description
The number of permutations less than or equal to a permutation in left weak order.
This is the same as the number of permutations less than or equal to the given permutation in right weak order.
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
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.