Abstract
Ciucu proved a simple product formula for the tiling number of a hexagon in which a chain of equilateral triangles of alternating orientations, called a `fern', has been removed from the center (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 central fern as in Ciucu's region, we remove two new ferns from two sides of the hexagon. Our result also implies a new `dual' of MacMahon's classical formula of boxed plane partitions, corresponding to the exterior of the union of three disjoint concave polygons obtained by turning 120 degrees after drawing each side.
Highlights
MacMahon’s classical theorem on plane partitions fitting in a given box is equivalent to the fact that the number of lozenge tilings of a centrally symmetric hexagon of side-lengths a, b, c, a, b, c on the triangular lattice is equal toH(a) H(b) H(c) H(a + b + c) P(a, b, c) := (1)H(a + b) H(b + c) H(c + a) where the hyperfactorial function H(n) is defined as H(n) := 0! · 1! · 2! · · · (n − 1)!
A lozenge is union of any two unit equilateral triangles sharing an edge; a lozenge tiling of a region on the triangular lattice is a covering of the region by lozenges without gaps or overlaps
We would like to investigate lozenge tilings of hexagons with certain ‘defects,’ and the most popular defect is the removal of a collection of one or more equilateral triangles
Summary
MacMahon’s classical theorem on plane partitions fitting in a given box (see [37], and [2,6,10,11,19,23,26,27,47,48] for more recent developments) is equivalent to the fact that the number of lozenge tilings of a centrally symmetric hexagon of side-lengths a, b, c, a, b, c (in clockwise order, starting from the north side) on the triangular lattice is equal to. Problem 2 on the list asks for a tiling formula for a quasi-regular the electronic journal of combinatorics 27(1) (2020), #P1.61 hexagon of side-lengths n, n+1, n, n+1, n, n+1 with the central unit triangle removed Our main theorems (Theorems 2–9) show that the numbers of tilings of the new regions are always given by a certain product of the tiling number of a cored hexagon, the tiling numbers of two dented semihexagons determined by the ferns, and a simple multiplicative factor.
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