191 research outputs found
Generating Random Elements of Finite Distributive Lattices
This survey article describes a method for choosing uniformly at random from
any finite set whose objects can be viewed as constituting a distributive
lattice. The method is based on ideas of the author and David Wilson for using
``coupling from the past'' to remove initialization bias from Monte Carlo
randomization. The article describes several applications to specific kinds of
combinatorial objects such as tilings, constrained lattice paths, and
alternating-sign matrices.Comment: 13 page
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
Three-player impartial games
Past efforts to classify impartial three-player combinatorial games (the
theories of Li and Straffin) have made various restrictive assumptions about
the rationality of one's opponents and the formation and behavior of
coalitions. One may instead adopt an agnostic attitude towards such issues, and
seek only to understand in what circumstances one player has a winning strategy
against the combined forces of the other two. By limiting ourselves to this
more modest theoretical objective, and by regarding two games as being
equivalent if they are interchangeable in all disjunctive sums,as far as
single-player winnability is concerned, we can obtain an interesting analogue
of Grundy values for three-player impartial games.Comment: 8 pages, 10 tables; to appear in Theoretical Computer Scienc
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