- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000881: Permutations ⟶ ℤ

Values

[1,0,1,0] => [1,2] => 0

[1,1,0,0] => [2,1] => 0

[1,0,1,0,1,0] => [1,2,3] => 0

[1,0,1,1,0,0] => [1,3,2] => 0

[1,1,0,0,1,0] => [2,1,3] => 0

[1,1,0,1,0,0] => [2,3,1] => 0

[1,1,1,0,0,0] => [3,2,1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,0,0,0,0] => [4,3,2,1] => 10

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 10

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => 7

[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => 37

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

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

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

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

[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => 19

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

[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => 7

[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => 12

[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => 89

[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 10

[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 37

[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => 89

[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => 250

[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 1386

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 10

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => 7

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => 37

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

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

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

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

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => 19

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

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => 7

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => 12

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => 89

[1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => 10

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$F_{2} = 2$

$F_{3} = 5$

$F_{4} = 9 + 3\ q + q^{2} + q^{10}$

$F_{5} = 14 + 7\ q + 7\ q^{2} + 2\ q^{6} + 2\ q^{7} + 2\ q^{10} + q^{12} + q^{19} + 2\ q^{37} + 2\ q^{89} + q^{250} + q^{1386}$

Description

The number of short 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 short braid move if they are obtained from each other by a modification of the form

$s_i s_j \leftrightarrow s_j s_i$ for

$|i-j| > 1$ as a consecutive subword of a reduced word.
For example, the two reduced words

$s_1s_3s_2$ and

$s_3s_1s_2$ for

$(1243) = (12)(34)(23) = (34)(12)(23)$
share an edge because they are obtained from each other by interchanging

$s_1s_3 \leftrightarrow s_3s_1$ .
This statistic counts the number of such short braid moves among all reduced words.

Map

to 312-avoiding permutation

Description

Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).