Tilings of hexagons with a removed triad of bowties

Simple Relationships Between Lozenge Tiling Functions of Related Regions

“…This is the second paper by this author and one of many in the field which identify simple relationships between related tiling functions which may not themselves be simple. Similar results have been given by Lai, Ciucu, Rohatgi, and Byun [3], [9], [10], [21], [22], [23]. In fact, in [24], Lai independently studies dented hexagons and dented half-hexagons with some equivalent or more general results to this paper, and in [14] Fulmek gives alternative proofs of some of these.…”

Problems in the Enumeration of Tilings

“…x,y,z (d, d ′ , e, e ′ , f, f ′ ) the new region. It has been shown that the ratio of tiling numbers of the above two regions is given by a simple product formula [19]. We would also like to emphasize that, in general, the tiling numbers of the regions R ∆…”

“…Several other examples of the shuffling phenomenon have been found. One of them is the shuffling phenomenon for a hexagon with a removed "triad of bowties" [19]. We consider the hexagon H of side-lengths…”

“…x,y,z (a, a ′ , b, b ′ , c, c ′ ) and R ∆ ′ x,y,z (d, d ′ , e, e ′ , f, f ′ ) are not given by simple product formulas (see Theorem 1 in [19]). See Figure 3 We have realized that the above instance of the shuffling phenomenon could be extended to the weighted case.…”

“…Byun generalized the shuffling phenomenon for a triad of bowties in [19]. He showed that the tiling number of a hexagon with holes on three crossing lines only changes by a simple multiplicative factor if we flip the central triangular hole and then translate these lines of holes (see Theorem 2.1 in [7]).…”