Séminaire Lotharingien de Combinatoire, B63e (2010), 13 pp.

Olivier Bernardi, Bertrand Duplantier and Philippe Nadeau

A Bijection Between Well-Labelled Positive Paths and Matchings

Abstract. A well-labelled positive path of size n is a pair (p,\sigma) made of a word p=p1p2...pn-1 on the alphabet {-1,0,+1} such that \sum_{i=1}^j pi >= 0 for all j=1,...,n-1, together with a permutation \sigma=\sigma1\sigma2...\sigman of {1,...,n} such that pi=-1 implies \sigmai<\sigmai+1, while pi=1 implies \sigmai>\sigmai+1. We establish a bijection between well-labelled positive paths of size n and matchings (i.e., fixed-point free involutions) on {1,...,2n}. This proves that the number of well-labelled positive paths is (2n-1)!!=(2n-1).(2n-3)...3.1.

Well-labelled positive paths appeared recently in the author's article "Partition function of a freely-jointed chain in a half-space" [in preparation] as a useful tool for studying a polytope \Pin related to the space of configurations of the freely-jointed chain (of length n) in a half-space. The polytope \Pin consists of points (x1,...,xn) in [-1,1]n such that \sum_{i=1}^j xi >= 0 for all j=1,...,n, and it was shown that well-labelled positive paths of size n are in bijection with a collection of subpolytopes partitioning \Pin. Given that the volume of each subpolytope is 1/n!, our results prove combinatorially that the volume of \Pin is (2n-1)!!/n!.

Our bijection has other enumerative corollaries in terms of up-down sequences of permutations. Indeed, by specialising our bijection, we prove that the number of permutations of size n such that each prefix has no more ascents than descents is [(n-1)!!]2 if n is even and n!!(n-2)!! if n is odd.

Received: November 11, 2009. Accepted: April 22, 2010. Final Version: April 23, 2010.

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