Exchange Symmetries in Motzkin Path and Bargraph Models of Copolymer Adsorption

In a previous work [26], by considering paths that are partially weighted, the generating function of Dyck paths was shown to possess a type of symmetry, called an exchange relation, derived from the exchange of a portion of the path between weighted and unweighted halves. This relation is particularly useful in solving for the generating functions of certain models of vertex-coloured Dyck paths; this is a directed model of copolymer adsorption, and in a particular case it is possible to find an asymptotic expression for the adsorption critical point of the model as a function of the colouring. In this paper we examine Motzkin path and partially directed walk models of the same adsorbing directed copolymer problem. These problems are an interesting generalisation of previous results since the colouring can be of either the edges, or the vertices, of the paths. We are able to find asymptotic expressions for the adsorption critical point in the Motzkin path model for both edge and verte...

• Type: Journal
• Year: 2002
• Where: COMBINATORICS
• Authors: E. J. Janse van Rensburg, A. Rechnitzer
• Keywords: Adsorption Critical Point, COMBINATORICS 2002, Dyck Paths, Generating Functions