
arXiv: 1803.02792
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.
graphical condensation, Exact enumeration problems, generating functions, Enumeration in graph theory, Tilings in \(2\) dimensions (aspects of discrete geometry), perfect matchings, lozenge tilings, 05A15, 05B45, plane partitions, dual graph, Combinatorial aspects of tessellation and tiling problems, Partitions of sets, FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO)
graphical condensation, Exact enumeration problems, generating functions, Enumeration in graph theory, Tilings in \(2\) dimensions (aspects of discrete geometry), perfect matchings, lozenge tilings, 05A15, 05B45, plane partitions, dual graph, Combinatorial aspects of tessellation and tiling problems, Partitions of sets, FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO)
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