Identifier
Values
[1,0] => [2,1] => [2,1] => 1
[1,0,1,0] => [3,1,2] => [1,3,2] => 1
[1,1,0,0] => [2,3,1] => [3,1,2] => 2
[1,0,1,0,1,0] => [4,1,2,3] => [1,2,4,3] => 1
[1,0,1,1,0,0] => [3,1,4,2] => [3,4,1,2] => 4
[1,1,0,0,1,0] => [2,4,1,3] => [1,3,4,2] => 2
[1,1,0,1,0,0] => [4,3,1,2] => [1,4,3,2] => 1
[1,1,1,0,0,0] => [2,3,4,1] => [4,1,2,3] => 3
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [2,4,5,1,3] => 5
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [3,1,4,5,2] => 4
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [2,1,5,4,3] => 2
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,5,1,2,4] => 5
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [4,2,5,1,3] => 4
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [1,4,2,5,3] => 3
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [1,2,5,4,3] => 1
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [4,5,2,1,3] => 5
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [1,3,4,5,2] => 3
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [1,4,5,3,2] => 3
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [1,5,2,4,3] => 2
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,2,3,4,6,5] => 1
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [2,3,5,6,1,4] => 6
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [2,4,1,5,6,3] => 5
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [2,3,1,6,5,4] => 3
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [2,4,6,1,3,5] => 6
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [3,1,4,2,6,5] => 4
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [4,5,2,6,1,3] => 7
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [2,1,5,3,6,4] => 4
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [2,1,3,6,5,4] => 2
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [3,5,6,2,1,4] => 7
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [3,1,4,5,6,2] => 5
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [4,1,5,6,3,2] => 6
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [2,1,6,3,5,4] => 3
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,6,1,2,4,5] => 6
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [1,3,2,4,6,5] => 2
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [2,4,5,6,1,3] => 7
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [4,2,1,5,6,3] => 4
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [2,4,1,6,5,3] => 4
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [4,2,6,1,3,5] => 5
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [1,4,2,3,6,5] => 3
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [2,5,4,6,1,3] => 6
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,2,5,3,6,4] => 3
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [1,2,3,6,4,5] => 2
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [2,5,6,3,1,4] => 6
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [4,1,5,3,6,2] => 5
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [4,1,3,6,5,2] => 3
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [4,1,6,2,5,3] => 5
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [4,6,2,1,3,5] => 6
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [1,3,4,2,6,5] => 3
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [5,2,3,6,1,4] => 4
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [1,4,5,2,6,3] => 5
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [1,4,2,6,5,3] => 3
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [5,3,6,2,1,4] => 6
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [1,5,2,3,6,4] => 4
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [1,2,4,6,5,3] => 2
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [1,2,6,5,4,3] => 2
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [5,6,1,3,2,4] => 7
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [1,3,4,5,6,2] => 4
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [1,4,5,6,3,2] => 5
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [1,4,6,2,5,3] => 4
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [1,6,2,3,5,4] => 3
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 1
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => [1,3,2,4,5,7,6] => 2
[1,1,0,1,0,0,1,0,1,0,1,0] => [7,3,1,2,4,5,6] => [1,4,2,3,5,7,6] => 3
[1,1,0,1,0,1,0,0,1,0,1,0] => [7,4,1,2,3,5,6] => [1,2,5,3,4,7,6] => 3
[1,1,0,1,0,1,0,1,0,0,1,0] => [5,7,1,2,3,4,6] => [1,2,3,4,6,7,5] => 2
[1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => 1
[1,1,1,0,0,0,1,0,1,0,1,0] => [2,3,7,1,4,5,6] => [1,3,4,2,5,7,6] => 3
[1,1,1,0,0,1,0,0,1,0,1,0] => [2,7,4,1,3,5,6] => [1,4,5,2,3,7,6] => 5
[1,1,1,0,0,1,0,1,0,0,1,0] => [2,7,5,1,3,4,6] => [1,4,2,6,3,7,5] => 5
[1,1,1,0,0,1,0,1,0,1,0,0] => [2,6,7,1,3,4,5] => [1,3,2,4,7,5,6] => 3
[1,1,1,0,1,0,0,1,0,0,1,0] => [7,3,5,1,2,4,6] => [1,2,4,6,3,7,5] => 4
[1,1,1,0,1,0,0,1,0,1,0,0] => [6,3,7,1,2,4,5] => [1,2,5,3,7,4,6] => 4
[1,1,1,0,1,0,1,0,0,0,1,0] => [7,5,4,1,2,3,6] => [1,2,6,5,3,7,4] => 4
[1,1,1,0,1,0,1,0,0,1,0,0] => [6,7,4,1,2,3,5] => [1,2,6,3,7,4,5] => 5
[1,1,1,0,1,0,1,0,1,0,0,0] => [6,7,5,1,2,3,4] => [1,2,3,7,6,4,5] => 3
[1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => [1,3,4,5,2,7,6] => 4
[1,1,1,1,0,0,0,1,0,0,1,0] => [2,3,7,5,1,4,6] => [1,4,5,6,2,7,3] => 7
[1,1,1,1,0,0,0,1,0,1,0,0] => [2,3,7,6,1,4,5] => [1,4,5,2,7,6,3] => 5
[1,1,1,1,0,0,1,0,0,0,1,0] => [2,7,4,5,1,3,6] => [1,4,6,2,3,7,5] => 6
[1,1,1,1,0,0,1,0,0,1,0,0] => [2,7,4,6,1,3,5] => [1,4,2,5,7,6,3] => 4
[1,1,1,1,0,1,0,0,0,1,0,0] => [7,3,4,6,1,2,5] => [1,2,4,5,7,6,3] => 3
[1,1,1,1,0,1,0,0,1,0,0,0] => [7,3,6,5,1,2,4] => [1,2,5,7,6,4,3] => 4
[1,1,1,1,0,1,0,1,0,0,0,0] => [7,6,4,5,1,2,3] => [1,2,7,3,6,5,4] => 3
[1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => [1,3,4,5,6,7,2] => 5
[1,1,1,1,1,0,0,0,0,1,0,0] => [2,3,4,7,6,1,5] => [1,4,5,6,7,3,2] => 7
[1,1,1,1,1,0,0,0,1,0,0,0] => [2,3,7,5,6,1,4] => [1,4,5,7,2,6,3] => 6
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Description
The reduced reflection length of the permutation.
Let $T$ be the set of reflections in a Coxeter group and let $\ell(w)$ be the usual length function. Then the reduced reflection length of $w$ is
$$\min\{r\in\mathbb N \mid w = t_1\cdots t_r,\quad t_1,\dots,t_r \in T,\quad \ell(w)=\sum \ell(t_i)\}.$$
In the case of the symmetric group, this is twice the depth St000029The depth of a permutation. minus the usual length St000018The number of inversions of a permutation..
Map
major-index to inversion-number bijection
Description
Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.