A Shuffling Theorem for Reflectively Symmetric Tilings

In arXiv:1905.08311, the author and Rohatgi proved a shuffling theorem for doubly-dented hexagons. In particular, we showed that shuffling removed unit triangles along a horizontal axis in a hexagon only changes the tiling number by a simple multiplicative factor. In this paper, we consider a similar phenomenon for a symmetry class of tilings, the reflectively symmetric tilings, of the doubly-dented hexagons. We also prove several shuffling theorems for halved hexagons. These theorems generalize a number of known results in the enumeration of halved hexagons.Comment: Second version: 21 page

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

MacMahon proved a simple product formula for the generating function of plane partitions fitting in a given box. The theorem implies a $q$-enumeration of lozenge tilings of a semi-regular hexagon on the triangular lattice. In this paper we generalize MacMahon's classical theorem by $q$-enumerating lozenge tilings of a new family of hexagons with four adjacent triangles removed from their boundary.Comment: 30 pages. Title is changed from "A q-enumeration of generalized plane partitions" to "A q-enumeration of lozenge tilings of a hexagon with four adjacent triangles removed from the boundary

A Shuffling Theorem for Centrally Symmetric Tilings

Rohatgi and the author recently proved a shuffling theorem for lozenge tilings of `doubly-dented hexagons' (arXiv:1905.08311). The theorem can be considered as a hybrid between two classical theorems in the enumeration of tilings: MacMahon's theorem about centrally symmetric hexagons and Cohn-Larsen-Prop's theorem about semihexagons with dents. In this paper, we consider a similar shuffling theorem for the centrally symmetric tilings of the doubly-dented hexagons. Our theorem also implies a conjecture posed by the author in arXiv:1803.02792 about the enumeration of centrally symmetric tilings of hexagons with three arrays of triangular holes. This enumeration, in turn, can be considered as a common generalization of (a tiling-equivalent version of) Stanley's enumeration of self-complementary plane partitions and Ciucu's work on symmetries of the shamrock structure. Moreover, our enumeration also confirms a recent conjecture posed by Ciucu in arXiv:1906.02951.Comment: Second version: 26 pages and many picture

Lozenge tilings of hexagons with central holes and dents

Ciucu showed that the number of lozenge tilings of a hexagon in which a chain of equilateral triangles of alternating orientations, called a `\emph{fern}', has been removed in the center is given by a simple product formula (Adv. Math. 2017). In this paper, we present a multi-parameter generalization of this work by giving an explicit tiling enumeration for a hexagon with three ferns removed, besides the middle fern located in the center as in Ciucu's region, we remove two additional ferns from two sides of the hexagon. Our result also implies a counterpart of MacMahon's classical formula of boxed plane partitions, corresponding the \emph{exterior} of the union of three disjoint concave polygons obtained by turning 120 degrees after drawing each side.Comment: Version 6: 52 pages. Version 6 is roughly the first half of Version 5, which also contained the material that is now arXiv:1905.07119. Version 6 contains minor expository change