Identifier
Values
0 => [2] => [1,1,0,0] => [2,3,1] => 2
1 => [1,1] => [1,0,1,0] => [3,1,2] => 2
00 => [3] => [1,1,1,0,0,0] => [2,3,4,1] => 3
01 => [2,1] => [1,1,0,0,1,0] => [2,4,1,3] => 4
10 => [1,2] => [1,0,1,1,0,0] => [3,1,4,2] => 4
11 => [1,1,1] => [1,0,1,0,1,0] => [4,1,2,3] => 3
000 => [4] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 4
001 => [3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 6
010 => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 7
011 => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 6
100 => [1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 6
101 => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 7
110 => [1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 6
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 4
=> [1] => [1,0] => [2,1] => 1
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Description
The reduced word complexity of a permutation.
For a permutation $\pi$, this is the smallest length of a word in simple transpositions that contains all reduced expressions of $\pi$.
For example, the permutation $[3,2,1] = (12)(23)(12) = (23)(12)(23)$ and the reduced word complexity is $4$ since the smallest words containing those two reduced words as subwords are $(12),(23),(12),(23)$ and also $(23),(12),(23),(12)$.
This statistic appears in [1, Question 6.1].
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.