Identifier
Values
[1] => {{1}} => [1] => [1] => 1
[1,2] => {{1},{2}} => [1,2] => [1,2] => 1
[2,1] => {{1,2}} => [2,1] => [2,1] => 2
[1,2,3] => {{1},{2},{3}} => [1,2,3] => [1,2,3] => 1
[1,3,2] => {{1},{2,3}} => [1,3,2] => [1,3,2] => 2
[2,1,3] => {{1,2},{3}} => [2,1,3] => [2,1,3] => 2
[2,3,1] => {{1,2,3}} => [2,3,1] => [2,3,1] => 3
[3,1,2] => {{1,2,3}} => [2,3,1] => [2,3,1] => 3
[3,2,1] => {{1,3},{2}} => [3,2,1] => [3,2,1] => 2
[1,2,3,4] => {{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => {{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => {{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => {{1},{2,3,4}} => [1,3,4,2] => [1,3,4,2] => 3
[1,4,2,3] => {{1},{2,3,4}} => [1,3,4,2] => [1,3,4,2] => 3
[1,4,3,2] => {{1},{2,4},{3}} => [1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => {{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => {{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => {{1,2,3},{4}} => [2,3,1,4] => [2,3,1,4] => 3
[2,3,4,1] => {{1,2,3,4}} => [2,3,4,1] => [2,3,4,1] => 4
[2,4,1,3] => {{1,2,3,4}} => [2,3,4,1] => [2,3,4,1] => 4
[2,4,3,1] => {{1,2,4},{3}} => [2,4,3,1] => [2,4,3,1] => 3
[3,1,2,4] => {{1,2,3},{4}} => [2,3,1,4] => [2,3,1,4] => 3
[3,1,4,2] => {{1,2,3,4}} => [2,3,4,1] => [2,3,4,1] => 4
[3,2,1,4] => {{1,3},{2},{4}} => [3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => {{1,3,4},{2}} => [3,2,4,1] => [3,2,4,1] => 3
[3,4,1,2] => {{1,3},{2,4}} => [3,4,1,2] => [3,4,1,2] => 2
[3,4,2,1] => {{1,2,3,4}} => [2,3,4,1] => [2,3,4,1] => 4
[4,1,2,3] => {{1,2,3,4}} => [2,3,4,1] => [2,3,4,1] => 4
[4,1,3,2] => {{1,2,4},{3}} => [2,4,3,1] => [2,4,3,1] => 3
[4,2,1,3] => {{1,3,4},{2}} => [3,2,4,1] => [3,2,4,1] => 3
[4,2,3,1] => {{1,4},{2},{3}} => [4,2,3,1] => [4,2,3,1] => 2
[4,3,1,2] => {{1,2,3,4}} => [2,3,4,1] => [2,3,4,1] => 4
[4,3,2,1] => {{1,4},{2,3}} => [4,3,2,1] => [4,3,2,1] => 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => {{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,4,3,5] => {{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,2,4,5,3] => {{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,4,5,3] => 3
[1,2,5,3,4] => {{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,4,5,3] => 3
[1,2,5,4,3] => {{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,3,2,5,4] => {{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => {{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,3,4,2,5] => 3
[1,3,4,5,2] => {{1},{2,3,4,5}} => [1,3,4,5,2] => [1,3,4,5,2] => 4
[1,3,5,2,4] => {{1},{2,3,4,5}} => [1,3,4,5,2] => [1,3,4,5,2] => 4
[1,3,5,4,2] => {{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,3,5,4,2] => 3
[1,4,2,3,5] => {{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,3,4,2,5] => 3
[1,4,2,5,3] => {{1},{2,3,4,5}} => [1,3,4,5,2] => [1,3,4,5,2] => 4
[1,4,3,2,5] => {{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,3,5,2] => {{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,4,3,5,2] => 3
[1,4,5,2,3] => {{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,4,5,2,3] => 2
[1,4,5,3,2] => {{1},{2,3,4,5}} => [1,3,4,5,2] => [1,3,4,5,2] => 4
[1,5,2,3,4] => {{1},{2,3,4,5}} => [1,3,4,5,2] => [1,3,4,5,2] => 4
[1,5,2,4,3] => {{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,3,5,4,2] => 3
[1,5,3,2,4] => {{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,4,3,5,2] => 3
[1,5,3,4,2] => {{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,5,4,2,3] => {{1},{2,3,4,5}} => [1,3,4,5,2] => [1,3,4,5,2] => 4
[1,5,4,3,2] => {{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,5,4,3,2] => 2
[2,1,3,4,5] => {{1,2},{3},{4},{5}} => [2,1,3,4,5] => [2,1,3,4,5] => 2
[2,1,3,5,4] => {{1,2},{3},{4,5}} => [2,1,3,5,4] => [2,1,3,5,4] => 2
[2,1,4,3,5] => {{1,2},{3,4},{5}} => [2,1,4,3,5] => [2,1,4,3,5] => 2
[2,1,4,5,3] => {{1,2},{3,4,5}} => [2,1,4,5,3] => [2,1,4,5,3] => 6
[2,1,5,3,4] => {{1,2},{3,4,5}} => [2,1,4,5,3] => [2,1,4,5,3] => 6
[2,1,5,4,3] => {{1,2},{3,5},{4}} => [2,1,5,4,3] => [2,1,5,4,3] => 2
[2,3,1,4,5] => {{1,2,3},{4},{5}} => [2,3,1,4,5] => [2,3,1,4,5] => 3
[2,3,1,5,4] => {{1,2,3},{4,5}} => [2,3,1,5,4] => [2,3,1,5,4] => 6
[2,3,4,1,5] => {{1,2,3,4},{5}} => [2,3,4,1,5] => [2,3,4,1,5] => 4
[2,3,4,5,1] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[2,3,5,1,4] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[2,3,5,4,1] => {{1,2,3,5},{4}} => [2,3,5,4,1] => [2,3,5,4,1] => 4
[2,4,1,3,5] => {{1,2,3,4},{5}} => [2,3,4,1,5] => [2,3,4,1,5] => 4
[2,4,1,5,3] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[2,4,3,1,5] => {{1,2,4},{3},{5}} => [2,4,3,1,5] => [2,4,3,1,5] => 3
[2,4,3,5,1] => {{1,2,4,5},{3}} => [2,4,3,5,1] => [2,4,3,5,1] => 4
[2,4,5,1,3] => {{1,2,4},{3,5}} => [2,4,5,1,3] => [2,4,5,1,3] => 6
[2,4,5,3,1] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[2,5,1,3,4] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[2,5,1,4,3] => {{1,2,3,5},{4}} => [2,3,5,4,1] => [2,3,5,4,1] => 4
[2,5,3,1,4] => {{1,2,4,5},{3}} => [2,4,3,5,1] => [2,4,3,5,1] => 4
[2,5,3,4,1] => {{1,2,5},{3},{4}} => [2,5,3,4,1] => [2,5,3,4,1] => 3
[2,5,4,1,3] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[2,5,4,3,1] => {{1,2,5},{3,4}} => [2,5,4,3,1] => [2,5,4,3,1] => 6
[3,1,2,4,5] => {{1,2,3},{4},{5}} => [2,3,1,4,5] => [2,3,1,4,5] => 3
[3,1,2,5,4] => {{1,2,3},{4,5}} => [2,3,1,5,4] => [2,3,1,5,4] => 6
[3,1,4,2,5] => {{1,2,3,4},{5}} => [2,3,4,1,5] => [2,3,4,1,5] => 4
[3,1,4,5,2] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[3,1,5,2,4] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[3,1,5,4,2] => {{1,2,3,5},{4}} => [2,3,5,4,1] => [2,3,5,4,1] => 4
[3,2,1,4,5] => {{1,3},{2},{4},{5}} => [3,2,1,4,5] => [3,2,1,4,5] => 2
[3,2,1,5,4] => {{1,3},{2},{4,5}} => [3,2,1,5,4] => [3,2,1,5,4] => 2
[3,2,4,1,5] => {{1,3,4},{2},{5}} => [3,2,4,1,5] => [3,2,4,1,5] => 3
[3,2,4,5,1] => {{1,3,4,5},{2}} => [3,2,4,5,1] => [3,2,4,5,1] => 4
[3,2,5,1,4] => {{1,3,4,5},{2}} => [3,2,4,5,1] => [3,2,4,5,1] => 4
[3,2,5,4,1] => {{1,3,5},{2},{4}} => [3,2,5,4,1] => [3,2,5,4,1] => 3
[3,4,1,2,5] => {{1,3},{2,4},{5}} => [3,4,1,2,5] => [3,4,1,2,5] => 2
[3,4,1,5,2] => {{1,3},{2,4,5}} => [3,4,1,5,2] => [3,4,1,5,2] => 6
[3,4,2,1,5] => {{1,2,3,4},{5}} => [2,3,4,1,5] => [2,3,4,1,5] => 4
[3,4,2,5,1] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[3,4,5,1,2] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[3,4,5,2,1] => {{1,3,5},{2,4}} => [3,4,5,2,1] => [3,4,5,2,1] => 6
[3,5,1,2,4] => {{1,3},{2,4,5}} => [3,4,1,5,2] => [3,4,1,5,2] => 6
[3,5,1,4,2] => {{1,3},{2,5},{4}} => [3,5,1,4,2] => [3,5,1,4,2] => 2
>>> Load all 149 entries. <<<
[3,5,2,1,4] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[3,5,2,4,1] => {{1,2,3,5},{4}} => [2,3,5,4,1] => [2,3,5,4,1] => 4
[3,5,4,1,2] => {{1,3,4},{2,5}} => [3,5,4,1,2] => [3,5,4,1,2] => 6
[3,5,4,2,1] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[4,1,2,3,5] => {{1,2,3,4},{5}} => [2,3,4,1,5] => [2,3,4,1,5] => 4
[4,1,2,5,3] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[4,1,3,2,5] => {{1,2,4},{3},{5}} => [2,4,3,1,5] => [2,4,3,1,5] => 3
[4,1,3,5,2] => {{1,2,4,5},{3}} => [2,4,3,5,1] => [2,4,3,5,1] => 4
[4,1,5,2,3] => {{1,2,4},{3,5}} => [2,4,5,1,3] => [2,4,5,1,3] => 6
[4,1,5,3,2] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[4,2,1,3,5] => {{1,3,4},{2},{5}} => [3,2,4,1,5] => [3,2,4,1,5] => 3
[4,2,1,5,3] => {{1,3,4,5},{2}} => [3,2,4,5,1] => [3,2,4,5,1] => 4
[4,2,3,1,5] => {{1,4},{2},{3},{5}} => [4,2,3,1,5] => [4,2,3,1,5] => 2
[4,2,3,5,1] => {{1,4,5},{2},{3}} => [4,2,3,5,1] => [4,2,3,5,1] => 3
[4,2,5,1,3] => {{1,4},{2},{3,5}} => [4,2,5,1,3] => [4,2,5,1,3] => 2
[4,2,5,3,1] => {{1,3,4,5},{2}} => [3,2,4,5,1] => [3,2,4,5,1] => 4
[4,3,1,2,5] => {{1,2,3,4},{5}} => [2,3,4,1,5] => [2,3,4,1,5] => 4
[4,3,1,5,2] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[4,3,2,1,5] => {{1,4},{2,3},{5}} => [4,3,2,1,5] => [4,3,2,1,5] => 2
[4,3,2,5,1] => {{1,4,5},{2,3}} => [4,3,2,5,1] => [4,3,2,5,1] => 6
[4,3,5,1,2] => {{1,4},{2,3,5}} => [4,3,5,1,2] => [4,3,5,1,2] => 6
[4,3,5,2,1] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[4,5,1,2,3] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[4,5,1,3,2] => {{1,3,4},{2,5}} => [3,5,4,1,2] => [3,5,4,1,2] => 6
[4,5,2,1,3] => {{1,4},{2,3,5}} => [4,3,5,1,2] => [4,3,5,1,2] => 6
[4,5,2,3,1] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[4,5,3,1,2] => {{1,4},{2,5},{3}} => [4,5,3,1,2] => [4,5,3,1,2] => 2
[4,5,3,2,1] => {{1,2,4,5},{3}} => [2,4,3,5,1] => [2,4,3,5,1] => 4
[5,1,2,3,4] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[5,1,2,4,3] => {{1,2,3,5},{4}} => [2,3,5,4,1] => [2,3,5,4,1] => 4
[5,1,3,2,4] => {{1,2,4,5},{3}} => [2,4,3,5,1] => [2,4,3,5,1] => 4
[5,1,3,4,2] => {{1,2,5},{3},{4}} => [2,5,3,4,1] => [2,5,3,4,1] => 3
[5,1,4,2,3] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[5,1,4,3,2] => {{1,2,5},{3,4}} => [2,5,4,3,1] => [2,5,4,3,1] => 6
[5,2,1,3,4] => {{1,3,4,5},{2}} => [3,2,4,5,1] => [3,2,4,5,1] => 4
[5,2,1,4,3] => {{1,3,5},{2},{4}} => [3,2,5,4,1] => [3,2,5,4,1] => 3
[5,2,3,1,4] => {{1,4,5},{2},{3}} => [4,2,3,5,1] => [4,2,3,5,1] => 3
[5,2,3,4,1] => {{1,5},{2},{3},{4}} => [5,2,3,4,1] => [5,2,3,4,1] => 2
[5,2,4,1,3] => {{1,3,4,5},{2}} => [3,2,4,5,1] => [3,2,4,5,1] => 4
[5,2,4,3,1] => {{1,5},{2},{3,4}} => [5,2,4,3,1] => [5,2,4,3,1] => 2
[5,3,1,2,4] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[5,3,1,4,2] => {{1,2,3,5},{4}} => [2,3,5,4,1] => [2,3,5,4,1] => 4
[5,3,2,1,4] => {{1,4,5},{2,3}} => [4,3,2,5,1] => [4,3,2,5,1] => 6
[5,3,4,1,2] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[5,4,1,2,3] => {{1,3,5},{2,4}} => [3,4,5,2,1] => [3,4,5,2,1] => 6
[5,4,1,3,2] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[5,4,2,1,3] => {{1,2,3,4,5}} => [2,3,4,5,1] => [2,3,4,5,1] => 5
[5,4,3,1,2] => {{1,2,4,5},{3}} => [2,4,3,5,1] => [2,4,3,5,1] => 4
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Description
The order of a signed permutation.
Map
to cycle type
Description
Let $\pi=c_1\dots c_r$ a permutation of size $n$ decomposed in its cyclic parts. The associated set partition of $[n]$ then is $S=S_1\cup\dots\cup S_r$ such that $S_i$ is the set of integers in the cycle $c_i$.
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].
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.