95 research outputs found
Proof of Blum's conjecture on hexagonal dungeons
Matt Blum conjectured that the number of tilings of the Hexagonal Dungeon of
sides (where ) is
(J. Propp, New Perspectives in
Geometric Combinatorics, Cambridge University Press, 1999). In this paper we
present a proof for this conjecture using Kuo's Graphical Condensation Theorem
(E. Kuo, Applications of Graphical Condensation for Enumerating Matchings and
Tilings, Theoretical Computer Science, 2004).Comment: 30 page
Applications of Graphical Condensation for Enumerating Matchings and Tilings
A technique called graphical condensation is used to prove various
combinatorial identities among numbers of (perfect) matchings of planar
bipartite graphs and tilings of regions. Graphical condensation involves
superimposing matchings of a graph onto matchings of a smaller subgraph, and
then re-partitioning the united matching (actually a multigraph) into matchings
of two other subgraphs, in one of two possible ways. This technique can be used
to enumerate perfect matchings of a wide variety of bipartite planar graphs.
Applications include domino tilings of Aztec diamonds and rectangles, diabolo
tilings of fortresses, plane partitions, and transpose complement plane
partitions.Comment: 25 pages; 21 figures Corrected typos; Updated references; Some text
revised, but content essentially the sam
Graphical condensation of plane graphs: a combinatorial approach
The method of graphical vertex-condensation for enumerating perfect matchings
of plane bipartite graph was found by Propp (Theoret. Comput. Sci. 303(2003),
267-301), and was generalized by Kuo (Theoret. Comput. Sci. 319 (2004), 29-57)
and Yan and Zhang (J. Combin. Theory Ser. A, 110(2005), 113-125). In this
paper, by a purely combinatorial method some explicit identities on graphical
vertex-condensation for enumerating perfect matchings of plane graphs (which do
not need to be bipartite) are obtained. As applications of our results, some
results on graphical edge-condensation for enumerating perfect matchings are
proved, and we count the sum of weights of perfect matchings of weighted Aztec
diamond.Comment: 13 pages, 5 figures. accepted by Theoretial Computer Scienc
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
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