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Electronic Journal of Combinatorics
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Lozenge Tilings of Hexagons with Central Holes and Dents

Lozenge tilings of hexagons with central holes and dents
Authors: Tri Lai;

Lozenge Tilings of Hexagons with Central Holes and Dents

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.

Keywords

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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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
1
Average
Average
Average
Green
gold