Distances on Lozenge Tilings

International audienceIn this paper, a structural property of the set of lozenge tilings of a 2n-gon is highlighted. We introduce a simple combinatorial value called Hamming-distance, which is a lower bound for the flipdistance (i.e. the number of necessary local transformations involving three lozenges) between two given tilings. It is here proven that, for n5, We show that there is some deficient pairs of tilings for which the flip connection needs more flips than the combinatorial lower bound indicates

A dual of MacMahon's theorem on plane partitions

A classical theorem of MacMahon states that the number of lozenge tilings of any centrally symmetric hexagon drawn on the triangular lattice is given by a beautifully simple product formula. In this paper we present a counterpart of this formula, corresponding to the {\it exterior} of a concave hexagon obtained by turning 120 degrees after drawing each side (MacMahon's hexagon is obtained by turning 60 degrees after each step).Comment: 22 page

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

Lozenge tilings of a halved hexagon with an array of triangles removed from the boundary

Proctor's work on staircase plane partitions yields an enumeration of lozenge tilings of a halved hexagon on the triangular lattice. Rohatgi recently extended this tiling enumeration to a halved hexagon with a triangle removed from the boundary. In this paper we prove a generalization of the results of Proctor and Rohatgi by enumerating lozenge tilings of a halved hexagon in which an array of adjacent triangles has been removed from the boundary.Comment: 28 pages. Third version: fixed several typo