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On the distance between the expressions of a permutation. (English) Zbl 1227.05013
Summary: We prove that the combinatorial distance between any two reduced expressions of a given permutation of $$\{ 1,\ldots ,n \}$$ in terms of transpositions lies in $$O(n^{4})$$. We prove that this bound is sharp, and, using a connection with the intersection numbers of certain curves in van Kampen diagrams, we give a practical criterion for proving that the derivations provided by the reversing algorithm of P. Dehornoy [“Groups with a complemented presentation,” J. Pure Appl. Algebra 116, No.1–3, 115–137 (1997; Zbl 0870.20023)] are optimal. We also show the existence of length $$\ell$$ expressions of different permutations whose reversing requires $$C\ell ^{4}$$ elementary steps.

##### MSC:
 05A05 Permutations, words, matrices 05E15 Combinatorial aspects of groups and algebras (MSC2010) 68R05 Combinatorics in computer science
##### Keywords:
combinatorial distance
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##### References:
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