Identifier
Values
[1] => [1,0] => [2,1] => 1
[1,1] => [1,0,1,0] => [3,1,2] => 1
[2] => [1,1,0,0] => [2,3,1] => 1
[1,1,1] => [1,0,1,0,1,0] => [4,1,2,3] => 1
[1,2] => [1,0,1,1,0,0] => [3,1,4,2] => 2
[2,1] => [1,1,0,0,1,0] => [2,4,1,3] => 2
[3] => [1,1,1,0,0,0] => [2,3,4,1] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 3
[1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 5
[1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 3
[2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 5
[3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 3
[4] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 9
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 6
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 9
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 16
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 11
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 4
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 11
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 16
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 9
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 6
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 9
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 4
[5] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => 14
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => 10
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => 5
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => 10
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [2,3,4,6,1,7,5] => 14
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => 5
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => 1
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Description
The number of reduced words for a permutation.
This is the number of ways to write a permutation as a minimal length product of simple transpositions. E.g., there are two reduced words for the permutation $[3,2,1]$, which are $(1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.