Twenty Open Problems in Enumeration of Matchings: Progress Report

Twenty Open Problems in Enumeration of Matchings

This document is an exposition of an assortment of open problems arising from the exact enumeration of (perfect) matchings of finite graphs. Roughly half have been solved at the time of this writing; see the document "Twenty Open Problems in Enumeration of Matchings: Progress Report" (also available from this server as math.CO/9801061). NOTE: This article has now been superseded by math.CO/9904150.

二十顶点配置与避免连续三个相等条目的格尔福德-茨特林模式之间的概率双射

A probabilistic bijection between twenty-vertex configurations with a free west boundary and Gelfand-Tsetlin patterns avoiding three equal entries in a row

本文研究了二十顶点配置与避免连续三个相等条目的格尔福德-茨特林模式之间的双射关系，通过构造概率双射来解释这两种枚举之间的联系，并推导出二十顶点配置的计数公式。

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 …

Enumeration of monochromatic three term arithmetic progressions in two-colorings of cyclic groups

One of the toughest problems in Ramsey theory is to determine the existence of monochromatic arithmetic progressions in groups whose elements have been colored. We study the harder problem to not only determine the existence of monochromatic arithmetic progressions, but to also count them. We reformulate the enumeration in real algebraic geometry and then use state of the art computational methods in semidefinite programming and representation theory to derive sharp, or an explicit constant from sharp, lower bounds for the cyclic group of any order.

Enumeration of monochromatic three term arithmetic progressions in two-colorings of any finite group

There are many extremely challenging problems about existence of monochromatic arithmetic progressions in colorings of groups. Many theorems hold only for abelian groups as results on non-abelian groups are often much more difficult to obtain. In this research project we do not only determine existence, but study the more general problem of counting them. We formulate the enumeration problem as a problem in real algebraic geometry and then use state of the art computational methods in semidefinite programming and representation theory to derive lower bounds for the number of monochromatic arit…

Enumeration of Splitting Subspaces over Finite Fields

We discuss an elementary, yet unsolved, problem of Niederreiter concerning the enumeration of a class of subspaces of finite dimensional vector spaces over finite fields. A short and self-contained account of some recent progress on this problem is included and some related problems are discussed.