On the distance between the expressions of a permutation

Details On the distance between the expressions of a permutation On the distance between the expressions of a permutation

2009-02-18

arxiv.org/abs/0902.3074v1

Mathematics

Combinatorics

Scientific paper

We prove that the combinatorial distance between any two reduced expressions of a given permutation of {1, ..., n} in terms of transpositions lies in O(n^4), a sharp bound. Using a connection with the intersection numbers of certain curves in van Kampen diagrams, we prove that this bound is sharp, and give a practical criterion for proving that the derivations provided by the reversing algorithm of [Dehornoy, JPAA 116 (1997) 115-197] are optimal. We also show the existence of length l expressions whose reversing requires C l^4 elementary steps.

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