156 research outputs found
Signed Lozenge Tilings
It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more general than hexagons. They are obtained by further removing upward-pointing triangles. We call the resulting shapes triangular regions. We establish signed versions of the latter three bijections for triangular regions. We first investigate the tileability of triangular regions by lozenges. Then we use perfect matchings and families of non-intersecting lattice paths to define two signs of a lozenge tiling. Using a new method that we call resolution of a puncture, we show that the two signs are in fact equivalent. As a consequence, we obtain the equality of determinants, up to sign, that enumerate signed perfect matchings and signed families of lattice paths of a triangular region, respectively. We also describe triangular regions, for which the signed enumerations agree with the unsigned enumerations
A factorization theorem for lozenge tilings of a hexagon with triangular holes
In this paper we present a combinatorial generalization of the fact that the
number of plane partitions that fit in a box is equal to
the number of such plane partitions that are symmetric, times the number of
such plane partitions for which the transpose is the same as the complement. We
use the equivalent phrasing of this identity in terms of symmetry classes of
lozenge tilings of a hexagon on the triangular lattice. Our generalization
consists of allowing the hexagon have certain symmetrically placed holes along
its horizontal symmetry axis. The special case when there are no holes can be
viewed as a new, simpler proof of the enumeration of symmetric plane
partitions.Comment: 20 page
Lozenge tilings with free boundaries
We study lozenge tilings of a domain with partially free boundary. In
particular, we consider a trapezoidal domain (half hexagon), s.t. the
horizontal lozenges on the long side can intersect it anywhere to protrude
halfway across. We show that the positions of the horizontal lozenges near the
opposite flat vertical boundary have the same joint distribution as the
eigenvalues from a Gaussian Unitary Ensemble (the GUE-corners/minors process).
We also prove the existence of a limit shape of the height function, which is
also a vertically symmetric plane partition. Both behaviors are shown to
coincide with those of the corresponding doubled fixed-boundary hexagonal
domain. We also consider domains where the different sides converge to
at different rates and recover again the GUE-corners process near the boundary.Comment: 27 pages, 4 figures; version 2-- typos fixed, improved proofs and
computations, incorporated referee's comments. To appear in Letters in
Mathematical Physic
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