Lozenge Tilings of a Halved Hexagon with an Array of Triangles Removed from the Boundary

Abstract

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 an arbitrary number of adjacent triangles has been removed from the boundary.

Lozenge Tilings of a Halved Hexagon with an Array of Triangles Removed from the Boundary, Part II

“…Similar to the case of Q ′ -type regions, we also define a weighted version K ′ (t) of the K(t) by assigning to each vertical lozenge on its western side a weight 1 2 (see Figure 2.1(f)). We adopt here the notations Q(t), Q ′ (t), K(t), and K ′ (t) from [19] for the numbers of tilings of the regions Q(t), Q ′ (t), K(t), and K ′ (t), respectively.…”

“…x + a 2 + a 4 Figure 1.2. Halved hexagons with an array of triangles removed from the nonstaircase boundary in [19].…”

“…From now on, we use the notations P a,b,c and P ′ a,b,c for the numbers of tilings of the regions P a,b,c and P ′ a,b,c , respectively. In the previous paper [19], the author generalized the tiling enumerations of a halved hexagon by Proctor and Rohatgi to a halved hexagon in which an array of an arbitrary number of adjacent triangles has been removed from the northeastern side (see Figure 1.2 for examples of the regions). In this paper, we investigate the other case when the array of triangles has been removed from the western side of the halved hexagon as shown in Figures 2.2, 2.3, and 2.4.…”

Lozenge Tilings of Hexagons with Central Holes and Dents

“…The main results in the series of papers [28,29,29] imply an explicit enumeration of a reflectively symmetric tilings of the Q ⊙ -type regions (i.e. tilings which are invariant under a refection over a vertical symmetry axis).…”

A q-enumeration of lozenge tilings of a hexagon with four adjacent triangles removed from the boundary

“…[Ci15], [Fu], [Ku06], [Sp], [YYZ], [YZ] for various aspects and generalizations of the method; and e.g. [CF16], [CF15], [CK], [CL], [KW], [La15a], [La15b], [La15c], [La16a], [La16b], [LM], [LMNT], [Ro15], [Ro16], [Zh] for recent applications of Kuo condensation.…”