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Identifier
Values
=>
[1,2]=>1 [2,1]=>1 [1,2,3]=>1 [1,3,2]=>1 [2,1,3]=>1 [2,3,1]=>1 [3,1,2]=>1 [3,2,1]=>1 [1,2,3,4]=>1 [1,2,4,3]=>1 [1,3,2,4]=>1 [1,3,4,2]=>1 [1,4,2,3]=>1 [1,4,3,2]=>1 [2,1,3,4]=>1 [2,1,4,3]=>2 [2,3,1,4]=>1 [2,3,4,1]=>1 [2,4,1,3]=>2 [2,4,3,1]=>2 [3,1,2,4]=>1 [3,1,4,2]=>2 [3,2,1,4]=>1 [3,2,4,1]=>2 [3,4,1,2]=>2 [3,4,2,1]=>3 [4,1,2,3]=>1 [4,1,3,2]=>2 [4,2,1,3]=>2 [4,2,3,1]=>4 [4,3,1,2]=>3 [4,3,2,1]=>8 [1,2,3,4,5]=>1 [1,2,3,5,4]=>1 [1,2,4,3,5]=>1 [1,2,4,5,3]=>1 [1,2,5,3,4]=>1 [1,2,5,4,3]=>1 [1,3,2,4,5]=>1 [1,3,2,5,4]=>2 [1,3,4,2,5]=>1 [1,3,4,5,2]=>1 [1,3,5,2,4]=>2 [1,3,5,4,2]=>2 [1,4,2,3,5]=>1 [1,4,2,5,3]=>2 [1,4,3,2,5]=>1 [1,4,3,5,2]=>2 [1,4,5,2,3]=>2 [1,4,5,3,2]=>3 [1,5,2,3,4]=>1 [1,5,2,4,3]=>2 [1,5,3,2,4]=>2 [1,5,3,4,2]=>4 [1,5,4,2,3]=>3 [1,5,4,3,2]=>8 [2,1,3,4,5]=>1 [2,1,3,5,4]=>2 [2,1,4,3,5]=>2 [2,1,4,5,3]=>3 [2,1,5,3,4]=>3 [2,1,5,4,3]=>6 [2,3,1,4,5]=>1 [2,3,1,5,4]=>3 [2,3,4,1,5]=>1 [2,3,4,5,1]=>1 [2,3,5,1,4]=>3 [2,3,5,4,1]=>3 [2,4,1,3,5]=>2 [2,4,1,5,3]=>5 [2,4,3,1,5]=>2 [2,4,3,5,1]=>3 [2,4,5,1,3]=>5 [2,4,5,3,1]=>6 [2,5,1,3,4]=>3 [2,5,1,4,3]=>9 [2,5,3,1,4]=>4 [2,5,3,4,1]=>7 [2,5,4,1,3]=>11 [2,5,4,3,1]=>20 [3,1,2,4,5]=>1 [3,1,2,5,4]=>3 [3,1,4,2,5]=>2 [3,1,4,5,2]=>3 [3,1,5,2,4]=>5 [3,1,5,4,2]=>9 [3,2,1,4,5]=>1 [3,2,1,5,4]=>6 [3,2,4,1,5]=>2 [3,2,4,5,1]=>3 [3,2,5,1,4]=>9 [3,2,5,4,1]=>14 [3,4,1,2,5]=>2 [3,4,1,5,2]=>5 [3,4,2,1,5]=>3 [3,4,2,5,1]=>6 [3,4,5,1,2]=>5 [3,4,5,2,1]=>9 [3,5,1,2,4]=>5 [3,5,1,4,2]=>14 [3,5,2,1,4]=>11 [3,5,2,4,1]=>25 [3,5,4,1,2]=>16 [3,5,4,2,1]=>42 [4,1,2,3,5]=>1 [4,1,2,5,3]=>3 [4,1,3,2,5]=>2 [4,1,3,5,2]=>4 [4,1,5,2,3]=>5 [4,1,5,3,2]=>11 [4,2,1,3,5]=>2 [4,2,1,5,3]=>9 [4,2,3,1,5]=>4 [4,2,3,5,1]=>7 [4,2,5,1,3]=>14 [4,2,5,3,1]=>25 [4,3,1,2,5]=>3 [4,3,1,5,2]=>11 [4,3,2,1,5]=>8 [4,3,2,5,1]=>20 [4,3,5,1,2]=>16 [4,3,5,2,1]=>42 [4,5,1,2,3]=>5 [4,5,1,3,2]=>16 [4,5,2,1,3]=>16 [4,5,2,3,1]=>41 [4,5,3,1,2]=>28 [4,5,3,2,1]=>99 [5,1,2,3,4]=>1 [5,1,2,4,3]=>3 [5,1,3,2,4]=>3 [5,1,3,4,2]=>7 [5,1,4,2,3]=>6 [5,1,4,3,2]=>20 [5,2,1,3,4]=>3 [5,2,1,4,3]=>14 [5,2,3,1,4]=>7 [5,2,3,4,1]=>14 [5,2,4,1,3]=>25 [5,2,4,3,1]=>56 [5,3,1,2,4]=>6 [5,3,1,4,2]=>25 [5,3,2,1,4]=>20 [5,3,2,4,1]=>56 [5,3,4,1,2]=>41 [5,3,4,2,1]=>133 [5,4,1,2,3]=>9 [5,4,1,3,2]=>42 [5,4,2,1,3]=>42 [5,4,2,3,1]=>133 [5,4,3,1,2]=>99 [5,4,3,2,1]=>432 [1,2,3,4,5,6]=>1 [1,2,3,4,6,5]=>1 [1,2,3,5,4,6]=>1 [1,2,3,5,6,4]=>1 [1,2,3,6,4,5]=>1 [1,2,3,6,5,4]=>1 [1,2,4,3,5,6]=>1 [1,2,4,3,6,5]=>2 [1,2,4,5,3,6]=>1 [1,2,4,5,6,3]=>1 [1,2,4,6,3,5]=>2 [1,2,4,6,5,3]=>2 [1,2,5,3,4,6]=>1 [1,2,5,3,6,4]=>2 [1,2,5,4,3,6]=>1 [1,2,5,4,6,3]=>2 [1,2,5,6,3,4]=>2 [1,2,5,6,4,3]=>3 [1,2,6,3,4,5]=>1 [1,2,6,3,5,4]=>2 [1,2,6,4,3,5]=>2 [1,2,6,4,5,3]=>4 [1,2,6,5,3,4]=>3 [1,2,6,5,4,3]=>8 [1,3,2,4,5,6]=>1 [1,3,2,4,6,5]=>2 [1,3,2,5,4,6]=>2 [1,3,2,5,6,4]=>3 [1,3,2,6,4,5]=>3 [1,3,2,6,5,4]=>6 [1,3,4,2,5,6]=>1 [1,3,4,2,6,5]=>3 [1,3,4,5,2,6]=>1 [1,3,4,5,6,2]=>1 [1,3,4,6,2,5]=>3 [1,3,4,6,5,2]=>3 [1,3,5,2,4,6]=>2 [1,3,5,2,6,4]=>5 [1,3,5,4,2,6]=>2 [1,3,5,4,6,2]=>3 [1,3,5,6,2,4]=>5 [1,3,5,6,4,2]=>6 [1,3,6,2,4,5]=>3 [1,3,6,2,5,4]=>9 [1,3,6,4,2,5]=>4 [1,3,6,4,5,2]=>7 [1,3,6,5,2,4]=>11 [1,3,6,5,4,2]=>20 [1,4,2,3,5,6]=>1 [1,4,2,3,6,5]=>3 [1,4,2,5,3,6]=>2 [1,4,2,5,6,3]=>3 [1,4,2,6,3,5]=>5 [1,4,2,6,5,3]=>9 [1,4,3,2,5,6]=>1 [1,4,3,2,6,5]=>6 [1,4,3,5,2,6]=>2 [1,4,3,5,6,2]=>3 [1,4,3,6,2,5]=>9 [1,4,3,6,5,2]=>14 [1,4,5,2,3,6]=>2 [1,4,5,2,6,3]=>5 [1,4,5,3,2,6]=>3 [1,4,5,3,6,2]=>6 [1,4,5,6,2,3]=>5 [1,4,5,6,3,2]=>9 [1,4,6,2,3,5]=>5 [1,4,6,2,5,3]=>14 [1,4,6,3,2,5]=>11 [1,4,6,3,5,2]=>25 [1,4,6,5,2,3]=>16 [1,4,6,5,3,2]=>42 [1,5,2,3,4,6]=>1 [1,5,2,3,6,4]=>3 [1,5,2,4,3,6]=>2 [1,5,2,4,6,3]=>4 [1,5,2,6,3,4]=>5 [1,5,2,6,4,3]=>11 [1,5,3,2,4,6]=>2 [1,5,3,2,6,4]=>9 [1,5,3,4,2,6]=>4 [1,5,3,4,6,2]=>7 [1,5,3,6,2,4]=>14 [1,5,3,6,4,2]=>25 [1,5,4,2,3,6]=>3 [1,5,4,2,6,3]=>11 [1,5,4,3,2,6]=>8 [1,5,4,3,6,2]=>20 [1,5,4,6,2,3]=>16 [1,5,4,6,3,2]=>42 [1,5,6,2,3,4]=>5 [1,5,6,2,4,3]=>16 [1,5,6,3,2,4]=>16 [1,5,6,3,4,2]=>41 [1,5,6,4,2,3]=>28 [1,5,6,4,3,2]=>99 [1,6,2,3,4,5]=>1 [1,6,2,3,5,4]=>3 [1,6,2,4,3,5]=>3 [1,6,2,4,5,3]=>7 [1,6,2,5,3,4]=>6 [1,6,2,5,4,3]=>20 [1,6,3,2,4,5]=>3 [1,6,3,2,5,4]=>14 [1,6,3,4,2,5]=>7 [1,6,3,4,5,2]=>14 [1,6,3,5,2,4]=>25 [1,6,3,5,4,2]=>56 [1,6,4,2,3,5]=>6 [1,6,4,2,5,3]=>25 [1,6,4,3,2,5]=>20 [1,6,4,3,5,2]=>56 [1,6,4,5,2,3]=>41 [1,6,4,5,3,2]=>133 [1,6,5,2,3,4]=>9 [1,6,5,2,4,3]=>42 [1,6,5,3,2,4]=>42 [1,6,5,3,4,2]=>133 [1,6,5,4,2,3]=>99 [1,6,5,4,3,2]=>432 [2,1,3,4,5,6]=>1 [2,1,3,4,6,5]=>2 [2,1,3,5,4,6]=>2 [2,1,3,5,6,4]=>3 [2,1,3,6,4,5]=>3 [2,1,3,6,5,4]=>6 [2,1,4,3,5,6]=>2 [2,1,4,3,6,5]=>6 [2,1,4,5,3,6]=>3 [2,1,4,5,6,3]=>4 [2,1,4,6,3,5]=>8 [2,1,4,6,5,3]=>12 [2,1,5,3,4,6]=>3 [2,1,5,3,6,4]=>8 [2,1,5,4,3,6]=>6 [2,1,5,4,6,3]=>12 [2,1,5,6,3,4]=>10 [2,1,5,6,4,3]=>22 [2,1,6,3,4,5]=>4 [2,1,6,3,5,4]=>12 [2,1,6,4,3,5]=>12 [2,1,6,4,5,3]=>28 [2,1,6,5,3,4]=>22 [2,1,6,5,4,3]=>72 [2,3,1,4,5,6]=>1 [2,3,1,4,6,5]=>3 [2,3,1,5,4,6]=>3 [2,3,1,5,6,4]=>6 [2,3,1,6,4,5]=>6 [2,3,1,6,5,4]=>17 [2,3,4,1,5,6]=>1 [2,3,4,1,6,5]=>4 [2,3,4,5,1,6]=>1 [2,3,4,5,6,1]=>1 [2,3,4,6,1,5]=>4 [2,3,4,6,5,1]=>4 [2,3,5,1,4,6]=>3 [2,3,5,1,6,4]=>9 [2,3,5,4,1,6]=>3 [2,3,5,4,6,1]=>4 [2,3,5,6,1,4]=>9 [2,3,5,6,4,1]=>10 [2,3,6,1,4,5]=>6 [2,3,6,1,5,4]=>23 [2,3,6,4,1,5]=>7 [2,3,6,4,5,1]=>11 [2,3,6,5,1,4]=>26 [2,3,6,5,4,1]=>40 [2,4,1,3,5,6]=>2 [2,4,1,3,6,5]=>8 [2,4,1,5,3,6]=>5 [2,4,1,5,6,3]=>9 [2,4,1,6,3,5]=>16 [2,4,1,6,5,3]=>35 [2,4,3,1,5,6]=>2 [2,4,3,1,6,5]=>12 [2,4,3,5,1,6]=>3 [2,4,3,5,6,1]=>4 [2,4,3,6,1,5]=>16 [2,4,3,6,5,1]=>22 [2,4,5,1,3,6]=>5 [2,4,5,1,6,3]=>14 [2,4,5,3,1,6]=>6 [2,4,5,3,6,1]=>10 [2,4,5,6,1,3]=>14 [2,4,5,6,3,1]=>19 [2,4,6,1,3,5]=>16 [2,4,6,1,5,3]=>51 [2,4,6,3,1,5]=>24 [2,4,6,3,5,1]=>46 [2,4,6,5,1,3]=>56 [2,4,6,5,3,1]=>101 [2,5,1,3,4,6]=>3 [2,5,1,3,6,4]=>11 [2,5,1,4,3,6]=>9 [2,5,1,4,6,3]=>21 [2,5,1,6,3,4]=>21 [2,5,1,6,4,3]=>61 [2,5,3,1,4,6]=>4 [2,5,3,1,6,4]=>21 [2,5,3,4,1,6]=>7 [2,5,3,4,6,1]=>11 [2,5,3,6,1,4]=>30 [2,5,3,6,4,1]=>47 [2,5,4,1,3,6]=>11 [2,5,4,1,6,3]=>42 [2,5,4,3,1,6]=>20 [2,5,4,3,6,1]=>40 [2,5,4,6,1,3]=>56 [2,5,4,6,3,1]=>104 [2,5,6,1,3,4]=>21 [2,5,6,1,4,3]=>82 [2,5,6,3,1,4]=>40 [2,5,6,3,4,1]=>87 [2,5,6,4,1,3]=>117 [2,5,6,4,3,1]=>276 [2,6,1,3,4,5]=>4 [2,6,1,3,5,4]=>16 [2,6,1,4,3,5]=>16 [2,6,1,4,5,3]=>44 [2,6,1,5,3,4]=>38 [2,6,1,5,4,3]=>150 [2,6,3,1,4,5]=>7 [2,6,3,1,5,4]=>40 [2,6,3,4,1,5]=>14 [2,6,3,4,5,1]=>25 [2,6,3,5,1,4]=>66 [2,6,3,5,4,1]=>124 [2,6,4,1,3,5]=>24 [2,6,4,1,5,3]=>107 [2,6,4,3,1,5]=>54 [2,6,4,3,5,1]=>121 [2,6,4,5,1,3]=>163 [2,6,4,5,3,1]=>361 [2,6,5,1,3,4]=>49 [2,6,5,1,4,3]=>259 [2,6,5,3,1,4]=>130 [2,6,5,3,4,1]=>331 [2,6,5,4,1,3]=>481 [2,6,5,4,3,1]=>1356 [3,1,2,4,5,6]=>1 [3,1,2,4,6,5]=>3 [3,1,2,5,4,6]=>3 [3,1,2,5,6,4]=>6 [3,1,2,6,4,5]=>6 [3,1,2,6,5,4]=>17 [3,1,4,2,5,6]=>2 [3,1,4,2,6,5]=>8 [3,1,4,5,2,6]=>3 [3,1,4,5,6,2]=>4 [3,1,4,6,2,5]=>11 [3,1,4,6,5,2]=>16 [3,1,5,2,4,6]=>5 [3,1,5,2,6,4]=>16 [3,1,5,4,2,6]=>9 [3,1,5,4,6,2]=>16 [3,1,5,6,2,4]=>21 [3,1,5,6,4,2]=>38 [3,1,6,2,4,5]=>9 [3,1,6,2,5,4]=>35 [3,1,6,4,2,5]=>21 [3,1,6,4,5,2]=>44 [3,1,6,5,2,4]=>61 [3,1,6,5,4,2]=>150 [3,2,1,4,5,6]=>1 [3,2,1,4,6,5]=>6 [3,2,1,5,4,6]=>6 [3,2,1,5,6,4]=>17 [3,2,1,6,4,5]=>17 [3,2,1,6,5,4]=>66 [3,2,4,1,5,6]=>2 [3,2,4,1,6,5]=>12 [3,2,4,5,1,6]=>3 [3,2,4,5,6,1]=>4 [3,2,4,6,1,5]=>16 [3,2,4,6,5,1]=>22 [3,2,5,1,4,6]=>9 [3,2,5,1,6,4]=>35 [3,2,5,4,1,6]=>14 [3,2,5,4,6,1]=>22 [3,2,5,6,1,4]=>44 [3,2,5,6,4,1]=>67 [3,2,6,1,4,5]=>23 [3,2,6,1,5,4]=>112 [3,2,6,4,1,5]=>40 [3,2,6,4,5,1]=>73 [3,2,6,5,1,4]=>165 [3,2,6,5,4,1]=>318 [3,4,1,2,5,6]=>2 [3,4,1,2,6,5]=>10 [3,4,1,5,2,6]=>5 [3,4,1,5,6,2]=>9 [3,4,1,6,2,5]=>21 [3,4,1,6,5,2]=>44 [3,4,2,1,5,6]=>3 [3,4,2,1,6,5]=>22 [3,4,2,5,1,6]=>6 [3,4,2,5,6,1]=>10 [3,4,2,6,1,5]=>38 [3,4,2,6,5,1]=>67 [3,4,5,1,2,6]=>5 [3,4,5,1,6,2]=>14 [3,4,5,2,1,6]=>9 [3,4,5,2,6,1]=>19 [3,4,5,6,1,2]=>14 [3,4,5,6,2,1]=>28 [3,4,6,1,2,5]=>21 [3,4,6,1,5,2]=>65 [3,4,6,2,1,5]=>49 [3,4,6,2,5,1]=>116 [3,4,6,5,1,2]=>70 [3,4,6,5,2,1]=>183 [3,5,1,2,4,6]=>5 [3,5,1,2,6,4]=>21 [3,5,1,4,2,6]=>14 [3,5,1,4,6,2]=>30 [3,5,1,6,2,4]=>42 [3,5,1,6,4,2]=>105 [3,5,2,1,4,6]=>11 [3,5,2,1,6,4]=>61 [3,5,2,4,1,6]=>25 [3,5,2,4,6,1]=>47 [3,5,2,6,1,4]=>105 [3,5,2,6,4,1]=>208 [3,5,4,1,2,6]=>16 [3,5,4,1,6,2]=>56 [3,5,4,2,1,6]=>42 [3,5,4,2,6,1]=>104 [3,5,4,6,1,2]=>70 [3,5,4,6,2,1]=>186 [3,5,6,1,2,4]=>42 [3,5,6,1,4,2]=>147 [3,5,6,2,1,4]=>126 [3,5,6,2,4,1]=>334 [3,5,6,4,1,2]=>187 [3,5,6,4,2,1]=>618 [3,6,1,2,4,5]=>9 [3,6,1,2,5,4]=>44 [3,6,1,4,2,5]=>30 [3,6,1,4,5,2]=>74 [3,6,1,5,2,4]=>105 [3,6,1,5,4,2]=>320 [3,6,2,1,4,5]=>26 [3,6,2,1,5,4]=>165 [3,6,2,4,1,5]=>66 [3,6,2,4,5,1]=>139 [3,6,2,5,1,4]=>330 [3,6,2,5,4,1]=>761
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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.
References
[1] Fishel, S., Milićević, E., Patrias, R., Tenner, B. E. Enumerations relating braid and commutation classes arXiv:1708.04372
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).connected_components() )
Created
Jul 03, 2017 at 16:58 by Christian Stump
Updated
Aug 24, 2017 at 23:36 by Martin Rubey