Tiling Enumeration of Hexagons with Off-Central Holes

Abstract

This paper is the sequel of the author's previous paper about tiling enumerations of the cored versions of a doubly-intruded hexagon (Electron. J. Combin. 2020), in which we generalized Ciucu's work about $F$-cored hexagons (Adv. Math. 2017). This paper provides an extensive list of thirty tiling enumerations of hexagons with three collinear chains of triangular holes.  Besides two chains of holes attaching to the boundary of the hexagon, we remove one more chain of triangles that is slightly off the center of the hexagon. Two of our enumerations imply two conjectures posed by Ciucu, Eisenkolbl, Krattenthaler, and Zare (J. Combin. Theory Ser. A 2001).