Tiling Generating Functions of Halved Hexagons and Quartered Hexagons

Lai, Tri; Rohatgi, Ranjan

doi:10.1007/s00026-021-00537-4

Cited by 5 publications

(3 citation statements)

References 22 publications

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“…We have observed a similar fact for a halved hexagon with defects [68]: the natural weight wt 2 does not give a nice q-enumeration, but the symmetric weight wt 3 does. Despite its very nice behavior, the study about the weight wt 3 is still extremely limited.…”

Section: Weighted Enumerations Of Lozenge Tilingsmentioning

confidence: 67%

“…It turns out the tiling numbers of the two regions are only different by a multiplicative factor, which is a perfect power of 2. As a nice q-formula for tiling generating function of the quartered hexagon has been found in [68], one would like to find such a q-formula for the quartered Aztec diamond and quartered Aztec rectangles.…”

Section: Weighted Enumerations Of Lozenge Tilingsmentioning

confidence: 99%

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Problems in the Enumeration of Tilings

Lai

2021

Preprint

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Enumeration of tilings is the mathematical study concerning the total number of coverings of regions by similar pieces without gaps or overlaps. Enumeration of tilings has become a vibrant subfield of combinatorics with connections and applications to diverse mathematical areas. In 1999, James Propp published his well-known list of 32 open problems in the field. The list has got much attention from experts around the world. After two decades, most of the problems on the list have been solved and generalized. In this paper, we propose a set of new tiling problems. This survey paper contributes to the Open Problems in Algebraic Combinatorics 2022 conference (OPAC 2022) at the University of Minnesota.

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Section: Weighted Enumerations Of Lozenge Tilingsmentioning

confidence: 67%

Section: Weighted Enumerations Of Lozenge Tilingsmentioning

confidence: 99%

Problems in the Enumeration of Tilings

Lai

2021

Preprint

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“…As there exist elegant q-analogs of MacMahon's tiling formula (also by MacMahon [8]), Cohn-Larsen-Propp's formula (in the language of column-strict plane partitions, see, e.g., [11, pp. 374-375]), and Proctor formula (see [7]), one should expect for a q-analog of our main theorem, at least for the…”

Section: Several Open Questionsmentioning

confidence: 71%

Tilted Halved Hexagons: Hexagons, Semi-hexagons, and Halved Hexagons Under One Roof

Lai

2020

Preprint

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We investigate a new family of regions that is the universal generalization of three wellknown region families in the field of enumeration of tilings: the quasi-regular hexagons, the semihexagons, and the halved hexagons. We prove a simple product formula for the number of tilings of these new regions. Our main result also yields the enumerations of two special classes of plane partitions with restricted parts.

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Lozenge Tilings of a Hexagon with a Horizontal Intrusion

Byun

2022

Ann. Comb.

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Tiling Generating Functions of Halved Hexagons and Quartered Hexagons

Cited by 5 publications

References 22 publications

Problems in the Enumeration of Tilings

Problems in the Enumeration of Tilings

Tilted Halved Hexagons: Hexagons, Semi-hexagons, and Halved Hexagons Under One Roof

Lozenge Tilings of a Hexagon with a Horizontal Intrusion

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