Domino shuffling for the Del Pezzo 3 lattice

Abstract

We present a version of the domino shuffling algorithm (due to Elkies, Kuperberg, Larsen and Propp) which works on a different lattice: the hexagonal lattice superimposed on its dual graph. We use our algorithm to count perfect matchings on a family of finite subgraphs of this lattice whose boundary conditions are compatible with our algorithm. In particular, we re-prove an enumerative theorem of Ciucu, as well as finding a related family of subgraphs which have 2^{(n+1)^2} perfect matchings. We also give three-variable generating functions for perfect matchings on both families of graphs, which encode certain statistics on the height functions of these graphs.

19 pages, 17 figures. V2: Changes to introduction and conclusion; references updated. V3: Correction to the statement of Theorem 7 and reworking of its proof

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Cited by in corpus (7)

• Colored BPS Pyramid Partition Functions, Quivers and Cluster Transformations
• Cluster Variables and Perfect Matchings of Subgraphs of the $d{P}_3$ Lattice
• Beyond Aztec Castles: Toric Cascades in the $d{P}_3$ Quiver
• Dungeons and Dragons: Combinatorics for the $d{P}_3$ Quiver
• Domino shuffling on Novak half-hexagons and Aztec half-diamonds
• A Generalization of Aztec Dragons
• Proof of a refinement of Blum's conjecture on hexagonal dungeons