Identifier
Values
[1] => [1] => 1
[1,2] => [1,2] => 1
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => 4
[3,1,2] => [3,2,1] => 4
[3,2,1] => [3,2,1] => 4
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,3,2] => 4
[1,4,2,3] => [1,4,3,2] => 4
[1,4,3,2] => [1,4,3,2] => 4
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [3,2,1,4] => 4
[2,3,4,1] => [4,2,3,1] => 64
[2,4,1,3] => [3,4,1,2] => 20
[2,4,3,1] => [4,3,2,1] => 168
[3,1,2,4] => [3,2,1,4] => 4
[3,1,4,2] => [4,2,3,1] => 64
[3,2,1,4] => [3,2,1,4] => 4
[3,2,4,1] => [4,2,3,1] => 64
[3,4,1,2] => [4,3,2,1] => 168
[3,4,2,1] => [4,3,2,1] => 168
[4,1,2,3] => [4,2,3,1] => 64
[4,1,3,2] => [4,2,3,1] => 64
[4,2,1,3] => [4,3,2,1] => 168
[4,2,3,1] => [4,3,2,1] => 168
[4,3,1,2] => [4,3,2,1] => 168
[4,3,2,1] => [4,3,2,1] => 168
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,5,4,3] => 4
[1,2,5,3,4] => [1,2,5,4,3] => 4
[1,2,5,4,3] => [1,2,5,4,3] => 4
[1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,4,3,2,5] => 4
[1,3,4,5,2] => [1,5,3,4,2] => 64
[1,3,5,2,4] => [1,4,5,2,3] => 20
[1,3,5,4,2] => [1,5,4,3,2] => 168
[1,4,2,3,5] => [1,4,3,2,5] => 4
[1,4,2,5,3] => [1,5,3,4,2] => 64
[1,4,3,2,5] => [1,4,3,2,5] => 4
[1,4,3,5,2] => [1,5,3,4,2] => 64
[1,4,5,2,3] => [1,5,4,3,2] => 168
[1,4,5,3,2] => [1,5,4,3,2] => 168
[1,5,2,3,4] => [1,5,3,4,2] => 64
[1,5,2,4,3] => [1,5,3,4,2] => 64
[1,5,3,2,4] => [1,5,4,3,2] => 168
[1,5,3,4,2] => [1,5,4,3,2] => 168
[1,5,4,2,3] => [1,5,4,3,2] => 168
[1,5,4,3,2] => [1,5,4,3,2] => 168
[2,1,3,4,5] => [2,1,3,4,5] => 1
[2,1,3,5,4] => [2,1,3,5,4] => 2
[2,1,4,3,5] => [2,1,4,3,5] => 2
[2,1,4,5,3] => [2,1,5,4,3] => 16
[2,1,5,3,4] => [2,1,5,4,3] => 16
[2,1,5,4,3] => [2,1,5,4,3] => 16
[2,3,1,4,5] => [3,2,1,4,5] => 4
[2,3,1,5,4] => [3,2,1,5,4] => 16
[2,3,4,1,5] => [4,2,3,1,5] => 64
[2,3,4,5,1] => [5,2,3,4,1] => 2176
[2,3,5,1,4] => [4,2,5,1,3] => 424
[2,3,5,4,1] => [5,2,4,3,1] => 8032
[2,4,1,3,5] => [3,4,1,2,5] => 20
[2,4,1,5,3] => [3,5,1,4,2] => 424
[2,4,3,1,5] => [4,3,2,1,5] => 168
[2,4,3,5,1] => [5,3,2,4,1] => 8032
[2,4,5,1,3] => [4,5,3,1,2] => 6720
[2,4,5,3,1] => [5,4,3,2,1] => 130560
[2,5,1,3,4] => [3,5,1,4,2] => 424
[2,5,1,4,3] => [3,5,1,4,2] => 424
[2,5,3,1,4] => [4,5,3,1,2] => 6720
[2,5,3,4,1] => [5,4,3,2,1] => 130560
[2,5,4,1,3] => [4,5,3,1,2] => 6720
[2,5,4,3,1] => [5,4,3,2,1] => 130560
[3,1,2,4,5] => [3,2,1,4,5] => 4
[3,1,2,5,4] => [3,2,1,5,4] => 16
[3,1,4,2,5] => [4,2,3,1,5] => 64
[3,1,4,5,2] => [5,2,3,4,1] => 2176
[3,1,5,2,4] => [4,2,5,1,3] => 424
[3,1,5,4,2] => [5,2,4,3,1] => 8032
[3,2,1,4,5] => [3,2,1,4,5] => 4
[3,2,1,5,4] => [3,2,1,5,4] => 16
[3,2,4,1,5] => [4,2,3,1,5] => 64
[3,2,4,5,1] => [5,2,3,4,1] => 2176
[3,2,5,1,4] => [4,2,5,1,3] => 424
[3,2,5,4,1] => [5,2,4,3,1] => 8032
[3,4,1,2,5] => [4,3,2,1,5] => 168
[3,4,1,5,2] => [5,3,2,4,1] => 8032
[3,4,2,1,5] => [4,3,2,1,5] => 168
[3,4,2,5,1] => [5,3,2,4,1] => 8032
[3,4,5,1,2] => [5,4,3,2,1] => 130560
[3,4,5,2,1] => [5,4,3,2,1] => 130560
[3,5,1,2,4] => [4,5,3,1,2] => 6720
[3,5,1,4,2] => [5,4,3,2,1] => 130560
>>> Load all 153 entries. <<<
[3,5,2,1,4] => [4,5,3,1,2] => 6720
[3,5,2,4,1] => [5,4,3,2,1] => 130560
[3,5,4,1,2] => [5,4,3,2,1] => 130560
[3,5,4,2,1] => [5,4,3,2,1] => 130560
[4,1,2,3,5] => [4,2,3,1,5] => 64
[4,1,2,5,3] => [5,2,3,4,1] => 2176
[4,1,3,2,5] => [4,2,3,1,5] => 64
[4,1,3,5,2] => [5,2,3,4,1] => 2176
[4,1,5,2,3] => [5,2,4,3,1] => 8032
[4,1,5,3,2] => [5,2,4,3,1] => 8032
[4,2,1,3,5] => [4,3,2,1,5] => 168
[4,2,1,5,3] => [5,3,2,4,1] => 8032
[4,2,3,1,5] => [4,3,2,1,5] => 168
[4,2,3,5,1] => [5,3,2,4,1] => 8032
[4,2,5,1,3] => [5,4,3,2,1] => 130560
[4,2,5,3,1] => [5,4,3,2,1] => 130560
[4,3,1,2,5] => [4,3,2,1,5] => 168
[4,3,1,5,2] => [5,3,2,4,1] => 8032
[4,3,2,1,5] => [4,3,2,1,5] => 168
[4,3,2,5,1] => [5,3,2,4,1] => 8032
[4,3,5,1,2] => [5,4,3,2,1] => 130560
[4,3,5,2,1] => [5,4,3,2,1] => 130560
[4,5,1,2,3] => [5,4,3,2,1] => 130560
[4,5,1,3,2] => [5,4,3,2,1] => 130560
[4,5,2,1,3] => [5,4,3,2,1] => 130560
[4,5,2,3,1] => [5,4,3,2,1] => 130560
[4,5,3,1,2] => [5,4,3,2,1] => 130560
[4,5,3,2,1] => [5,4,3,2,1] => 130560
[5,1,2,3,4] => [5,2,3,4,1] => 2176
[5,1,2,4,3] => [5,2,3,4,1] => 2176
[5,1,3,2,4] => [5,2,4,3,1] => 8032
[5,1,3,4,2] => [5,2,4,3,1] => 8032
[5,1,4,2,3] => [5,2,4,3,1] => 8032
[5,1,4,3,2] => [5,2,4,3,1] => 8032
[5,2,1,3,4] => [5,3,2,4,1] => 8032
[5,2,1,4,3] => [5,3,2,4,1] => 8032
[5,2,3,1,4] => [5,4,3,2,1] => 130560
[5,2,3,4,1] => [5,4,3,2,1] => 130560
[5,2,4,1,3] => [5,4,3,2,1] => 130560
[5,2,4,3,1] => [5,4,3,2,1] => 130560
[5,3,1,2,4] => [5,4,3,2,1] => 130560
[5,3,1,4,2] => [5,4,3,2,1] => 130560
[5,3,2,1,4] => [5,4,3,2,1] => 130560
[5,3,2,4,1] => [5,4,3,2,1] => 130560
[5,3,4,1,2] => [5,4,3,2,1] => 130560
[5,3,4,2,1] => [5,4,3,2,1] => 130560
[5,4,1,2,3] => [5,4,3,2,1] => 130560
[5,4,1,3,2] => [5,4,3,2,1] => 130560
[5,4,2,1,3] => [5,4,3,2,1] => 130560
[5,4,2,3,1] => [5,4,3,2,1] => 130560
[5,4,3,1,2] => [5,4,3,2,1] => 130560
[5,4,3,2,1] => [5,4,3,2,1] => 130560
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Description
The number of Bruhat factorizations of a permutation.
This is the number of factorizations $\pi = t_1 \cdots t_\ell$ for transpositions $\{ t_i \mid 1 \leq i \leq \ell\}$ such that the number of inversions of $t_1 \cdots t_i$ equals $i$ for all $1 \leq i \leq \ell$.
Map
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.