Geodesic
Propp's benzels and Lai's nearly symmetric hexagons with holes
The electronic journal of combinatorics, Tome 32 (2025) no. 4
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In this paper we present a new version of the second author's factorization theorem for perfect matchings of symmetric graphs. We then use our result to solve four open problems of Propp on the enumeration of trimer tilings on the hexagonal lattice. As another application, we obtain a semi-factorization result for the number of lozenge tilings of a large class of hexagonal regions with holes (obtained by starting with an arbitrary symmetric hexagon with holes, and translating all the holes one unit lattice segment in the same direction). This in turn leads to the solution of two open problems posed by Lai, to an extension of a result due to Fulmek and Krattenthaler, which result in exact enumeration formulas for some new families of hexagonal regions with holes. Our result also allows us to find new, simpler proofs (and in one case, a new, simpler form) of some formulas due to Krattenthaler for the number of perfect matchings of Aztec rectangles with unit holes along a lattice diagonal.
DOI : 10.37236/13772
Classification : 05A15, 05A19, 05B45, 05C70
Mots-clés : lattice paths, domino tilings, Aztec diamond, chessboard, Aztec rectangle, perfect matching, Schur function identities 
@article{10_37236_13772,
     author = {Seok Hyun  Byun and Mihai Ciucu and Yi-Lin Lee},
     title = {Propp's benzels and {Lai's} nearly symmetric hexagons with holes},
     journal = {The electronic journal of combinatorics},
     year = {2025},
     volume = {32},
     number = {4},
     doi = {10.37236/13772},
     zbl = {8120121},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/13772/}
}
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Seok Hyun  Byun; Mihai Ciucu; Yi-Lin Lee. Propp's benzels and Lai's nearly symmetric hexagons with holes. The electronic journal of combinatorics, Tome 32 (2025) no. 4. doi: 10.37236/13772

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