On the distance between the expressions of a permutation

Marc Autord (1) , Patrick Dehornoy (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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.

Mots clés: permutation; transposition; reduced expression; combinatorial distance; van Kampen diagram; subword reversing; braid relation

Domaines: Combinatoire [math.CO]

Marc Autord, Patrick Dehornoy. On the distance between the expressions of a permutation. 2009. ⟨hal-00362372⟩