Identifier
Values
{{1}} => [1] => [1] => 0
{{1,2}} => [2,1] => [2,1] => 1
{{1},{2}} => [1,2] => [1,2] => 0
{{1,2,3}} => [2,3,1] => [2,3,1] => 2
{{1,2},{3}} => [2,1,3] => [2,1,3] => 1
{{1,3},{2}} => [3,2,1] => [3,2,1] => 2
{{1},{2,3}} => [1,3,2] => [1,3,2] => 1
{{1},{2},{3}} => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}} => [2,3,4,1] => [2,3,4,1] => 3
{{1,2,3},{4}} => [2,3,1,4] => [2,3,1,4] => 2
{{1,2,4},{3}} => [2,4,3,1] => [2,4,3,1] => 3
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => 2
{{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => 1
{{1,3,4},{2}} => [3,2,4,1] => [3,2,4,1] => 3
{{1,3},{2,4}} => [3,4,1,2] => [3,4,1,2] => 4
{{1,3},{2},{4}} => [3,2,1,4] => [3,2,1,4] => 2
{{1,4},{2,3}} => [4,3,2,1] => [4,3,2,1] => 4
{{1},{2,3,4}} => [1,3,4,2] => [1,3,4,2] => 2
{{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}} => [4,2,3,1] => [4,2,3,1] => 3
{{1},{2,4},{3}} => [1,4,3,2] => [1,4,3,2] => 2
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3,4,5}} => [1,3,4,5,2] => [1,3,4,5,2] => 3
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,3,4,2,5] => 2
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,3,5,4,2] => 3
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,3,2,5,4] => 2
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,4,3,5,2] => 3
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,4,5,2,3] => 4
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,4,3,2,5] => 2
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,5,4,3,2] => 4
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,4,5,3] => 2
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,5,3,4,2] => 3
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,5,4,3] => 2
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4,5] => 0
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Description
The depth of a signed permutation.
The depth of a positive root is its rank in the root poset. The depth of an element of a Coxeter group is the minimal sum of depths for any representation as product of reflections.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.