Identifier
- St000881: Permutations ⟶ ℤ
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 1
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 2
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 4
[4,3,1,2] => 2
[4,3,2,1] => 10
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 1
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 0
[1,4,2,5,3] => 1
[1,4,3,2,5] => 0
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
[1,4,5,3,2] => 2
[1,5,2,3,4] => 0
[1,5,2,4,3] => 1
[1,5,3,2,4] => 1
[1,5,3,4,2] => 4
[1,5,4,2,3] => 2
[1,5,4,3,2] => 10
[2,1,3,4,5] => 0
[2,1,3,5,4] => 1
[2,1,4,3,5] => 1
[2,1,4,5,3] => 2
[2,1,5,3,4] => 2
[2,1,5,4,3] => 6
[2,3,1,4,5] => 0
[2,3,1,5,4] => 2
[2,3,4,1,5] => 0
[2,3,4,5,1] => 0
[2,3,5,1,4] => 2
[2,3,5,4,1] => 2
[2,4,1,3,5] => 1
[2,4,1,5,3] => 5
[2,4,3,1,5] => 1
[2,4,3,5,1] => 2
[2,4,5,1,3] => 5
[2,4,5,3,1] => 7
[2,5,1,3,4] => 2
[2,5,1,4,3] => 11
[2,5,3,1,4] => 4
[2,5,3,4,1] => 9
[2,5,4,1,3] => 16
[2,5,4,3,1] => 37
[3,1,2,4,5] => 0
[3,1,2,5,4] => 2
[3,1,4,2,5] => 1
[3,1,4,5,2] => 2
[3,1,5,2,4] => 5
[3,1,5,4,2] => 11
[3,2,1,4,5] => 0
[3,2,1,5,4] => 6
[3,2,4,1,5] => 1
[3,2,4,5,1] => 2
[3,2,5,1,4] => 11
[3,2,5,4,1] => 19
[3,4,1,2,5] => 1
[3,4,1,5,2] => 5
[3,4,2,1,5] => 2
[3,4,2,5,1] => 7
[3,4,5,1,2] => 5
[3,4,5,2,1] => 12
[3,5,1,2,4] => 5
[3,5,1,4,2] => 21
[3,5,2,1,4] => 16
>>> Load all 500 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
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.
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.
Code
def short_braid_move_graph(pi):
V = [ tuple(w) for w in pi.reduced_words() ]
is_edge = lambda w1,w2: w1 != w2 and any( w1[:i] == w2[:i] and w1[i+2:] == w2[i+2:] for i in range(len(w1)-1) )
return Graph([V,is_edge])
def statistic(pi):
return len( short_braid_move_graph(pi).edges() )
Created
Jul 03, 2017 at 16:58 by Christian Stump
Updated
Jul 03, 2017 at 16:58 by Christian Stump
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!