56 research outputs found
Lambda-determinants and domino-tilings
Consider the -by- matrix with for satisfying and for all
other , consisting of a central diamond of 1's surrounded by 0's. When , the -determinant of the matrix (as introduced by Robbins
and Rumsey) is not well-defined. However, if we replace the 0's by 's, we
get a matrix whose -determinant is well-defined and is a polynomial in
and . The limit of this polynomial as is a polynomial in
whose value at is the number of domino tilings of a
-by- square.Comment: 4 pages; to appear in a special issue of Advances in Applied
Mathematics honoring David P. Robbin
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
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 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
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