Enumeration of Matchings: Problems and Progress

This document is built around a list of thirty-two problems in enumeration of matchings, the first twenty of which were presented in a lecture at MSRI in the fall of 1996. I begin with a capsule history of the topic of enumeration of matchings. The twenty original problems, with commentary, comprise the bulk of the article. I give an account of the progress that has been made on these problems as of this writing, and include pointers to both the printed and on-line literature; roughly half of the original twenty problems were solved by participants in the MSRI Workshop on Combinatorics, their students, and others, between 1996 and 1999. The article concludes with a dozen new open problems. (Note: This article supersedes math.CO/9801060 and math.CO/9801061.)Comment: 1+37 pages; to appear in "New Perspectives in Geometric Combinatorics" (ed. by Billera, Bjorner, Green, Simeon, and Stanley), Mathematical Science Research Institute publication #37, Cambridge University Press, 199

A Robinson-Schensted algorithm for a class of partial orders

Let P be a finite partial order which does not contain an induced subposet isomorphic with 3 + 1, and let G be the incomparability graph of P . Gasharov has shown that the chromatic symmetric function XG has nonnegative coefficients when expanded in terms of Schur functions; his proof uses the dual Jacobi-Trudi identity and a sign-reversing involution to interpret these coefficients as numbers of P -tableau. This suggests the possibility of a direct bijective proof of this result, generalizing the Robinson-Schensted correspondence. We provide such an algorithm here under the additional hypothesis that P does not contain an induced subposet isomorphic with fx ? a ! b ! c ? yg. 0. Introduction. An apt subtitle for this paper would be "a long complicated argument for a special case of a more general theorem which has a short elegant proof". As such, we must not only present our result clearly, but also explain why "doing it the hard way" is interesting. First, we present the background ..