- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000879: Permutations ⟶ ℤ

Values

=> [2] => [1,1,0,0] => [2,3,1] => 0
=> [1,1] => [1,0,1,0] => [3,1,2] => 0
=> [3] => [1,1,1,0,0,0] => [2,3,4,1] => 0
=> [2,1] => [1,1,0,0,1,0] => [2,4,1,3] => 0
=> [1,2] => [1,0,1,1,0,0] => [3,1,4,2] => 0
=> [1,1,1] => [1,0,1,0,1,0] => [4,1,2,3] => 0
=> [4] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 0
=> [3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 0
=> [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 0
=> [2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 0
=> [1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 0
=> [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
=> [1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
=> [1,1,1,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
=> [5] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 0
=> [4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 0
=> [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 0
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 0
=> [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 0
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 0
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 0
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 0
=> [1,4] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 0
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 0
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 0
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 0
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 0
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 0
 => [1] => [1,0] => [2,1] => 0

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$F_{1} = 2$

$F_{2} = 4$

$F_{3} = 8$

Description

The number of long braid edges in the graph of braid moves of a permutation.
Given a permutation

$\pi$ , let

$\operatorname{Red}(\pi)$ denote the set of reduced words for

$\pi$ in terms of simple transpositions

$s_i = (i,i+1)$ . We now say that two reduced words are connected by a long braid move if they are obtained from each other by a modification of the form

$s_i s_{i+1} s_i \leftrightarrow s_{i+1} s_i s_{i+1}$ as a consecutive subword of a reduced word.
For example, the two reduced words

$s_1s_3s_2s_3$ and

$s_1s_2s_3s_2$ for

$(124) = (12)(34)(23)(34) = (12)(23)(34)(23)$
share an edge because they are obtained from each other by interchanging

$s_3s_2s_3 \leftrightarrow s_3s_2s_3$ .
This statistic counts the number of such short braid moves among all reduced words.

Map

Ringel

Description

The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.

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

bounce path

Description

The bounce path determined by an integer composition.