Identifier
Values
[1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => 2
[2,1] => [1,2] => [1,2] => 2
[1,2,3] => [1,2,3] => [1,2,3] => 6
[1,3,2] => [1,2,3] => [1,2,3] => 6
[2,1,3] => [1,2,3] => [1,2,3] => 6
[2,3,1] => [1,2,3] => [1,2,3] => 6
[3,1,2] => [1,3,2] => [1,2,3] => 6
[3,2,1] => [1,3,2] => [1,2,3] => 6
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 20
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 20
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 20
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 20
[1,4,2,3] => [1,2,4,3] => [1,2,3,4] => 20
[1,4,3,2] => [1,2,4,3] => [1,2,3,4] => 20
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 20
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 20
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 20
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 20
[2,4,1,3] => [1,2,4,3] => [1,2,3,4] => 20
[2,4,3,1] => [1,2,4,3] => [1,2,3,4] => 20
[3,1,2,4] => [1,3,2,4] => [1,2,3,4] => 20
[3,1,4,2] => [1,3,4,2] => [1,2,3,4] => 20
[3,2,1,4] => [1,3,2,4] => [1,2,3,4] => 20
[3,2,4,1] => [1,3,4,2] => [1,2,3,4] => 20
[3,4,1,2] => [1,3,2,4] => [1,2,3,4] => 20
[3,4,2,1] => [1,3,2,4] => [1,2,3,4] => 20
[4,1,2,3] => [1,4,3,2] => [1,2,4,3] => 12
[4,1,3,2] => [1,4,2,3] => [1,2,4,3] => 12
[4,2,1,3] => [1,4,3,2] => [1,2,4,3] => 12
[4,2,3,1] => [1,4,2,3] => [1,2,4,3] => 12
[4,3,1,2] => [1,4,2,3] => [1,2,4,3] => 12
[4,3,2,1] => [1,4,2,3] => [1,2,4,3] => 12
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 66
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 66
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 66
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 66
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,4,5] => 66
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,4,5] => 66
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 66
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 66
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 66
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 66
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,4,5] => 66
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,4,5] => 66
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => 66
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,3,4,5] => 66
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,3,4,5] => 66
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,3,4,5] => 66
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,3,4,5] => 66
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,3,4,5] => 66
[1,5,2,3,4] => [1,2,5,4,3] => [1,2,3,5,4] => 36
[1,5,2,4,3] => [1,2,5,3,4] => [1,2,3,5,4] => 36
[1,5,3,2,4] => [1,2,5,4,3] => [1,2,3,5,4] => 36
[1,5,3,4,2] => [1,2,5,3,4] => [1,2,3,5,4] => 36
[1,5,4,2,3] => [1,2,5,3,4] => [1,2,3,5,4] => 36
[1,5,4,3,2] => [1,2,5,3,4] => [1,2,3,5,4] => 36
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 66
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 66
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 66
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 66
[2,1,5,3,4] => [1,2,3,5,4] => [1,2,3,4,5] => 66
[2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,4,5] => 66
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 66
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 66
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 66
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 66
[2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,4,5] => 66
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,4,5] => 66
[2,4,1,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => 66
[2,4,1,5,3] => [1,2,4,5,3] => [1,2,3,4,5] => 66
[2,4,3,1,5] => [1,2,4,3,5] => [1,2,3,4,5] => 66
[2,4,3,5,1] => [1,2,4,5,3] => [1,2,3,4,5] => 66
[2,4,5,1,3] => [1,2,4,3,5] => [1,2,3,4,5] => 66
[2,4,5,3,1] => [1,2,4,3,5] => [1,2,3,4,5] => 66
[2,5,1,3,4] => [1,2,5,4,3] => [1,2,3,5,4] => 36
[2,5,1,4,3] => [1,2,5,3,4] => [1,2,3,5,4] => 36
[2,5,3,1,4] => [1,2,5,4,3] => [1,2,3,5,4] => 36
[2,5,3,4,1] => [1,2,5,3,4] => [1,2,3,5,4] => 36
[2,5,4,1,3] => [1,2,5,3,4] => [1,2,3,5,4] => 36
[2,5,4,3,1] => [1,2,5,3,4] => [1,2,3,5,4] => 36
[3,1,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => 66
[3,1,2,5,4] => [1,3,2,4,5] => [1,2,3,4,5] => 66
[3,1,4,2,5] => [1,3,4,2,5] => [1,2,3,4,5] => 66
[3,1,4,5,2] => [1,3,4,5,2] => [1,2,3,4,5] => 66
[3,1,5,2,4] => [1,3,5,4,2] => [1,2,3,5,4] => 36
[3,1,5,4,2] => [1,3,5,2,4] => [1,2,3,5,4] => 36
[3,2,1,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => 66
[3,2,1,5,4] => [1,3,2,4,5] => [1,2,3,4,5] => 66
[3,2,4,1,5] => [1,3,4,2,5] => [1,2,3,4,5] => 66
[3,2,4,5,1] => [1,3,4,5,2] => [1,2,3,4,5] => 66
[3,2,5,1,4] => [1,3,5,4,2] => [1,2,3,5,4] => 36
[3,2,5,4,1] => [1,3,5,2,4] => [1,2,3,5,4] => 36
[3,4,1,2,5] => [1,3,2,4,5] => [1,2,3,4,5] => 66
[3,4,1,5,2] => [1,3,2,4,5] => [1,2,3,4,5] => 66
[3,4,2,1,5] => [1,3,2,4,5] => [1,2,3,4,5] => 66
[3,4,2,5,1] => [1,3,2,4,5] => [1,2,3,4,5] => 66
[3,4,5,1,2] => [1,3,5,2,4] => [1,2,3,5,4] => 36
[3,4,5,2,1] => [1,3,5,2,4] => [1,2,3,5,4] => 36
[3,5,1,2,4] => [1,3,2,5,4] => [1,2,3,4,5] => 66
[3,5,1,4,2] => [1,3,2,5,4] => [1,2,3,4,5] => 66
>>> Load all 153 entries. <<<
[3,5,2,1,4] => [1,3,2,5,4] => [1,2,3,4,5] => 66
[3,5,2,4,1] => [1,3,2,5,4] => [1,2,3,4,5] => 66
[3,5,4,1,2] => [1,3,4,2,5] => [1,2,3,4,5] => 66
[3,5,4,2,1] => [1,3,4,2,5] => [1,2,3,4,5] => 66
[4,1,2,3,5] => [1,4,3,2,5] => [1,2,4,3,5] => 48
[4,1,2,5,3] => [1,4,5,3,2] => [1,2,4,3,5] => 48
[4,1,3,2,5] => [1,4,2,3,5] => [1,2,4,3,5] => 48
[4,1,3,5,2] => [1,4,5,2,3] => [1,2,4,3,5] => 48
[4,1,5,2,3] => [1,4,2,3,5] => [1,2,4,3,5] => 48
[4,1,5,3,2] => [1,4,3,5,2] => [1,2,4,5,3] => 31
[4,2,1,3,5] => [1,4,3,2,5] => [1,2,4,3,5] => 48
[4,2,1,5,3] => [1,4,5,3,2] => [1,2,4,3,5] => 48
[4,2,3,1,5] => [1,4,2,3,5] => [1,2,4,3,5] => 48
[4,2,3,5,1] => [1,4,5,2,3] => [1,2,4,3,5] => 48
[4,2,5,1,3] => [1,4,2,3,5] => [1,2,4,3,5] => 48
[4,2,5,3,1] => [1,4,3,5,2] => [1,2,4,5,3] => 31
[4,3,1,2,5] => [1,4,2,3,5] => [1,2,4,3,5] => 48
[4,3,1,5,2] => [1,4,5,2,3] => [1,2,4,3,5] => 48
[4,3,2,1,5] => [1,4,2,3,5] => [1,2,4,3,5] => 48
[4,3,2,5,1] => [1,4,5,2,3] => [1,2,4,3,5] => 48
[4,3,5,1,2] => [1,4,2,3,5] => [1,2,4,3,5] => 48
[4,3,5,2,1] => [1,4,2,3,5] => [1,2,4,3,5] => 48
[4,5,1,2,3] => [1,4,2,5,3] => [1,2,4,5,3] => 31
[4,5,1,3,2] => [1,4,3,2,5] => [1,2,4,3,5] => 48
[4,5,2,1,3] => [1,4,2,5,3] => [1,2,4,5,3] => 31
[4,5,2,3,1] => [1,4,3,2,5] => [1,2,4,3,5] => 48
[4,5,3,1,2] => [1,4,2,5,3] => [1,2,4,5,3] => 31
[4,5,3,2,1] => [1,4,2,5,3] => [1,2,4,5,3] => 31
[5,1,2,3,4] => [1,5,4,3,2] => [1,2,5,3,4] => 31
[5,1,2,4,3] => [1,5,3,2,4] => [1,2,5,4,3] => 12
[5,1,3,2,4] => [1,5,4,2,3] => [1,2,5,3,4] => 31
[5,1,3,4,2] => [1,5,2,3,4] => [1,2,5,4,3] => 12
[5,1,4,2,3] => [1,5,3,4,2] => [1,2,5,3,4] => 31
[5,1,4,3,2] => [1,5,2,3,4] => [1,2,5,4,3] => 12
[5,2,1,3,4] => [1,5,4,3,2] => [1,2,5,3,4] => 31
[5,2,1,4,3] => [1,5,3,2,4] => [1,2,5,4,3] => 12
[5,2,3,1,4] => [1,5,4,2,3] => [1,2,5,3,4] => 31
[5,2,3,4,1] => [1,5,2,3,4] => [1,2,5,4,3] => 12
[5,2,4,1,3] => [1,5,3,4,2] => [1,2,5,3,4] => 31
[5,2,4,3,1] => [1,5,2,3,4] => [1,2,5,4,3] => 12
[5,3,1,2,4] => [1,5,4,2,3] => [1,2,5,3,4] => 31
[5,3,1,4,2] => [1,5,2,3,4] => [1,2,5,4,3] => 12
[5,3,2,1,4] => [1,5,4,2,3] => [1,2,5,3,4] => 31
[5,3,2,4,1] => [1,5,2,3,4] => [1,2,5,4,3] => 12
[5,3,4,1,2] => [1,5,2,3,4] => [1,2,5,4,3] => 12
[5,3,4,2,1] => [1,5,2,3,4] => [1,2,5,4,3] => 12
[5,4,1,2,3] => [1,5,3,2,4] => [1,2,5,4,3] => 12
[5,4,1,3,2] => [1,5,2,4,3] => [1,2,5,3,4] => 31
[5,4,2,1,3] => [1,5,3,2,4] => [1,2,5,4,3] => 12
[5,4,2,3,1] => [1,5,2,4,3] => [1,2,5,3,4] => 31
[5,4,3,1,2] => [1,5,2,4,3] => [1,2,5,3,4] => 31
[5,4,3,2,1] => [1,5,2,4,3] => [1,2,5,3,4] => 31
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Description
The number of parabolic double cosets with minimal element being the given permutation.
For $w \in S_n$, this is
$$\big| W_I \tau W_J\ :\ \tau \in S_n,\ I,J \subseteq S,\ w = \min\{W_I \tau W_J\}\big|$$
where $S$ is the set of simple transpositions, $W_K$ is the parabolic subgroup generated by $K \subseteq S$, and $\min\{W_I \tau W_J\}$ is the unique minimal element in weak order in the double coset $W_I \tau W_J$.
[1] contains a combinatorial description of these parabolic double cosets which can be used to compute this statistic.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.