263 research outputs found
A generalization of Aztec diamond theorem, part I
We generalize Aztec diamond theorem (N. Elkies, G. Kuperberg, M. Larsen, and
J. Propp, Alternating-sign matrices and domino tilings, Journal Algebraic
Combinatoric, 1992) by showing that the numbers of tilings of a certain family
of regions in the square lattice with southwest-to-northeast diagonals drawn in
are given by powers of 2. We present a proof for the generalization by using a
bijection between domino tilings and non-intersecting lattice paths.Comment: 18 page
A Generalization of Aztec Diamond Theorem, Part II
The author gave a proof of a generalization of the Aztec diamond theorem for
a family of -vertex regions on the square lattice with
southwest-to-northeast diagonals drawn in (Electron. J. Combin., 2014) by using
a bijection between tilings and non-intersecting lattice paths. In this paper,
we use Kuo graphical condensation to give a new proof.Comment: 11 pages and 7 figure
Enumeration of Hybrid Domino-Lozenge Tilings
We solve and generalize an open problem posted by James Propp (Problem 16 in
New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999)
on the number of tilings of quasi-hexagonal regions on the square lattice with
every third diagonal drawn in. We also obtain a generalization of Douglas'
Theorem on the number of tilings of a family of regions of the square lattice
with every second diagonal drawn in.Comment: 35 pages, 31 figure
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
Local statistics for random domino tilings of the Aztec diamond
We prove an asymptotic formula for the probability that, if one chooses a
domino tiling of a large Aztec diamond at random according to the uniform
distribution on such tilings, the tiling will contain a domino covering a given
pair of adjacent lattice squares. This formula quantifies the effect of the
diamond's boundary conditions on the behavior of typical tilings; in addition,
it yields a new proof of the arctic circle theorem of Jockusch, Propp, and
Shor. Our approach is to use the saddle point method to estimate certain
weighted sums of squares of Krawtchouk polynomials (whose relevance to domino
tilings is demonstrated elsewhere), and to combine these estimates with some
exponential sum bounds to deduce our final result. This approach generalizes
straightforwardly to the case in which the probability distribution on the set
of tilings incorporates bias favoring horizontal over vertical tiles or vice
versa. We also prove a fairly general large deviation estimate for domino
tilings of simply-connected planar regions that implies that some of our
results on Aztec diamonds apply to many other similar regions as well.Comment: 42 pages, 7 figure
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