Mihai Ciucu

Enumeration of Lozenge Tilings of Hexagons with a Central Triangular Hole

We deal with unweighted and weighted enumerations of lozenge tilings of a hexagon with side lengths a, b+m, c, a+m, b, c+m, where an equilateral triangle of side length m has been removed from the center. We give closed formulas for the plain enumeration and for a certain (?1)-enumeration of these lozenge tilings. In the case that a=b=c, we also provide closed formulas for certain weighted enumerations of those lozenge tilings that are cyclically symmetric. For m=0, the latter formulas specialize to statements about weighted enumerations of cyclically symmetric plane partitions. One such specialization gives a proof of a conjecture of Stembridge on a certain weighted count of cyclically symmetric plane partitions. The tools employed in our proofs are nonstandard applications of the theory of nonintersecting lattice paths and determinant evaluations. In particular, we evaluate the determinants det0?i, j?n?1(??ij+(m+i+jj)), where ? is any 6th root of unity. These determinant evaluations are variations of a famous result due to Andrews (1979, Invent. Math.53, 193?225), which corresponds to ?=1.

A Complementation Theorem for Perfect Matchings of Graphs Having a Cellular Completion

A cellular graph is a graph whose edges can be partitioned into 4-cycles (called cells) so that each vertex is contained in at most two cells. We present a “Complementation Theorem” for the number of matchings of certain subgraphs of cellular graphs. This generalizes the main result of M. Ciucu (J. Algebraic Combin.5(1996), 87?103). As applications of the Complementation Theorem we obtain a new proof of Stanleys multivariate version of the Aztec diamond theorem, a weighted generalization of a result of Knuth (J. Algebraic Combin.6(1997), 253?257) concerning spanning trees of Aztec diamond graphs, a combinatorial proof of Yangs enumeration (“Three Enumeration Problems Concerning Aztec Diamonds,” Ph.D. thesis, M.I.T., 1991) of matchings of fortress graphs and direct proofs for certain identities of Jockusch and Propp.

Dimer packings with gaps and electrostatics

Fisher and Stephenson conjectured in 1963 that the correlation function (defined by dimer packings) of two unit holes on the square lattice is rotationally invariant in the limit of large separation between the holes. We consider the same problem on the hexagonal lattice, extend it to an arbitrary finite collection of holes, and present an explicit conjectural answer. In recent work we managed to prove this conjecture in two fairly general cases. The quantity giving the answer can be regarded as the exponential of the negative of the two-dimensional electrostatic energy of a system of charges naturally associated with the holes. We further develop this analogy to electrostatics by presenting two different natural ways to define a field in our setup, and showing that both lead to the electric field, in the limit of large separations between the holes. For one of the fields, this is also stated as a limit shape theorem for random surfaces, with the continuum limit being a sum of helicoids. We conclude by explaining the relationship of our results to previous results in the physics literature on spin correlations in the Ising model.

A random tiling model for two dimensional electrostatics

Part A. A Random Tiling Model for Two Dimensional Electrostatics: Introduction Definitions, statement of results and physical interpretation Reduction to boundary-influenced correlations A simple product formula for correlations along the boundary A

Perfect Matchings and Perfect Powers

(2m+2n)

The scaling limit of the correlation of holes on the triangular lattice with periodic boundary conditions

-fold sum for

The Number of Centered Lozenge Tilings of a Symmetric Hexagon

\omega_b

A dual of MacMahon’s theorem on plane partitions

Separation of the

The number of spanning trees of plane graphs with reflective symmetry

(2m+2n)

The Interaction of a Gap with a Free Boundary in a Two Dimensional Dimer System

-fold sum for