Identifier
Values
[1,0,1,0] => [1,2] => 1
[1,1,0,0] => [2,1] => 1
[1,0,1,0,1,0] => [1,2,3] => 1
[1,0,1,1,0,0] => [1,3,2] => 1
[1,1,0,0,1,0] => [2,1,3] => 1
[1,1,0,1,0,0] => [2,3,1] => 1
[1,1,1,0,0,0] => [3,2,1] => 1
[1,0,1,0,1,0,1,0] => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0] => [1,3,4,2] => 1
[1,0,1,1,1,0,0,0] => [1,4,3,2] => 1
[1,1,0,0,1,0,1,0] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0] => [2,3,1,4] => 1
[1,1,0,1,0,1,0,0] => [2,3,4,1] => 1
[1,1,0,1,1,0,0,0] => [2,4,3,1] => 2
[1,1,1,0,0,0,1,0] => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0] => [3,2,4,1] => 2
[1,1,1,0,1,0,0,0] => [3,4,2,1] => 3
[1,1,1,1,0,0,0,0] => [4,3,2,1] => 8
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 1
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 1
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => 1
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 1
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => 2
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 1
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => 2
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => 3
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 8
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 3
[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] => 1
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 3
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 1
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 1
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => 3
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => 2
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => 3
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => 6
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => 20
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 1
[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] => 2
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => 3
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => 14
[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => 3
[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => 6
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => 9
[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => 42
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 8
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 20
[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => 42
[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => 99
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 432
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => 3
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 8
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => 3
[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] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => 3
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => 3
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => 6
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => 20
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => 1
[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] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => 3
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => 14
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => 6
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => 9
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => 42
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => 8
>>> Load all 174 entries. <<<
[1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => 20
[1,0,1,1,1,1,0,0,1,0,0,0] => [1,5,4,6,3,2] => 42
[1,0,1,1,1,1,0,1,0,0,0,0] => [1,5,6,4,3,2] => 99
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => 432
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => 1
[1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => 2
[1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => 2
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,1,3,5,6,4] => 3
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,5,4] => 6
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => 2
[1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => 6
[1,1,0,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => 3
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,1,4,5,6,3] => 4
[1,1,0,0,1,1,0,1,1,0,0,0] => [2,1,4,6,5,3] => 12
[1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,4,3,6] => 6
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,1,5,4,6,3] => 12
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,1,5,6,4,3] => 22
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,5,4,3] => 72
[1,1,0,1,0,0,1,0,1,0,1,0] => [2,3,1,4,5,6] => 1
[1,1,0,1,0,0,1,0,1,1,0,0] => [2,3,1,4,6,5] => 3
[1,1,0,1,0,0,1,1,0,0,1,0] => [2,3,1,5,4,6] => 3
[1,1,0,1,0,0,1,1,0,1,0,0] => [2,3,1,5,6,4] => 6
[1,1,0,1,0,0,1,1,1,0,0,0] => [2,3,1,6,5,4] => 17
[1,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,1,5,6] => 1
[1,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,1,6,5] => 4
[1,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,1,6] => 1
[1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => 1
[1,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,6,5,1] => 4
[1,1,0,1,0,1,1,0,0,0,1,0] => [2,3,5,4,1,6] => 3
[1,1,0,1,0,1,1,0,0,1,0,0] => [2,3,5,4,6,1] => 4
[1,1,0,1,0,1,1,0,1,0,0,0] => [2,3,5,6,4,1] => 10
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,3,6,5,4,1] => 40
[1,1,0,1,1,0,0,0,1,0,1,0] => [2,4,3,1,5,6] => 2
[1,1,0,1,1,0,0,0,1,1,0,0] => [2,4,3,1,6,5] => 12
[1,1,0,1,1,0,0,1,0,0,1,0] => [2,4,3,5,1,6] => 3
[1,1,0,1,1,0,0,1,0,1,0,0] => [2,4,3,5,6,1] => 4
[1,1,0,1,1,0,0,1,1,0,0,0] => [2,4,3,6,5,1] => 22
[1,1,0,1,1,0,1,0,0,0,1,0] => [2,4,5,3,1,6] => 6
[1,1,0,1,1,0,1,0,0,1,0,0] => [2,4,5,3,6,1] => 10
[1,1,0,1,1,0,1,0,1,0,0,0] => [2,4,5,6,3,1] => 19
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,4,6,5,3,1] => 101
[1,1,0,1,1,1,0,0,0,0,1,0] => [2,5,4,3,1,6] => 20
[1,1,0,1,1,1,0,0,0,1,0,0] => [2,5,4,3,6,1] => 40
[1,1,0,1,1,1,0,0,1,0,0,0] => [2,5,4,6,3,1] => 104
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,5,6,4,3,1] => 276
[1,1,0,1,1,1,1,0,0,0,0,0] => [2,6,5,4,3,1] => 1356
[1,1,1,0,0,0,1,0,1,0,1,0] => [3,2,1,4,5,6] => 1
[1,1,1,0,0,0,1,0,1,1,0,0] => [3,2,1,4,6,5] => 6
[1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => 6
[1,1,1,0,0,0,1,1,0,1,0,0] => [3,2,1,5,6,4] => 17
[1,1,1,0,0,0,1,1,1,0,0,0] => [3,2,1,6,5,4] => 66
[1,1,1,0,0,1,0,0,1,0,1,0] => [3,2,4,1,5,6] => 2
[1,1,1,0,0,1,0,0,1,1,0,0] => [3,2,4,1,6,5] => 12
[1,1,1,0,0,1,0,1,0,0,1,0] => [3,2,4,5,1,6] => 3
[1,1,1,0,0,1,0,1,0,1,0,0] => [3,2,4,5,6,1] => 4
[1,1,1,0,0,1,0,1,1,0,0,0] => [3,2,4,6,5,1] => 22
[1,1,1,0,0,1,1,0,0,0,1,0] => [3,2,5,4,1,6] => 14
[1,1,1,0,0,1,1,0,0,1,0,0] => [3,2,5,4,6,1] => 22
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,5,6,4,1] => 67
[1,1,1,0,0,1,1,1,0,0,0,0] => [3,2,6,5,4,1] => 318
[1,1,1,0,1,0,0,0,1,0,1,0] => [3,4,2,1,5,6] => 3
[1,1,1,0,1,0,0,0,1,1,0,0] => [3,4,2,1,6,5] => 22
[1,1,1,0,1,0,0,1,0,0,1,0] => [3,4,2,5,1,6] => 6
[1,1,1,0,1,0,0,1,0,1,0,0] => [3,4,2,5,6,1] => 10
[1,1,1,0,1,0,0,1,1,0,0,0] => [3,4,2,6,5,1] => 67
[1,1,1,0,1,0,1,0,0,0,1,0] => [3,4,5,2,1,6] => 9
[1,1,1,0,1,0,1,0,0,1,0,0] => [3,4,5,2,6,1] => 19
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,2,1] => 28
[1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,6,5,2,1] => 183
[1,1,1,0,1,1,0,0,0,0,1,0] => [3,5,4,2,1,6] => 42
[1,1,1,0,1,1,0,0,0,1,0,0] => [3,5,4,2,6,1] => 104
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,5,4,6,2,1] => 186
[1,1,1,0,1,1,0,1,0,0,0,0] => [3,5,6,4,2,1] => 618
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Description
The number of connected components 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 connected components of such long 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.).