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Identifier

St000879: Permutations ⟶ ℤ

Values

[1] => 0

[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] => 1

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[4,3,2,1] => 8

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

[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] => 2

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

[1,5,4,3,2] => 8

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,5,4,3,1] => 15

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 5 + q$

$F_{4} = 14 + 6\ q + 3\ q^{2} + q^{8}$

$F_{5} = 42 + 17\ q + 16\ q^{2} + 8\ q^{3} + 10\ q^{5} + 3\ q^{6} + 2\ q^{8} + 4\ q^{11} + 4\ q^{15} + 3\ q^{16} + 4\ q^{30} + 2\ q^{36} + 2\ q^{77} + 2\ q^{91} + q^{384}$

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.

Code

def long_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+3:] == w2[i+3:] and w1[i] != w2[i] and w1[i+2] != w2[i+2] and w1[i] == w1[i+2]for i in range(len(w1)-2) )
    return Graph([V,is_edge])

def statistic(pi):
    return len( long_braid_move_graph(pi).edges() )

Created

Jul 03, 2017 at 16:58 by Christian Stump

Updated

Apr 01, 2018 at 22:25 by Martin Rubey