On the distance between the expressions of a permutation

Abstract. 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(n4), 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 expressions whose reversing requires C 4 elementary steps. This paper is about the various ways of expressing a permutation as a product of transpositions and the complexity of transforming one such expression into another. We consider both the absolute complexity ("combinatorial distance"), which deals with the minimal possible number of steps, and the more specific complexity ("reversing complexity"), which arises when one uses subword reversing, a certain prescribed strategy ...

• Type: Journal
• Year: 2010
• Where: EJC
• Authors: Patrick Dehornoy, Marc Autord
• Keywords: Combinatorial Distance, EJC 2007, EJC 2010, Sharp Bound, Van Kampen