Dungeons and Dragons: Combinatorics for the dP₃ Quiver

In this paper, we utilize the machinery of cluster algebras, quiver mutations, and brane tilings to study a variety of historical enumerative combinatorics questions all under one roof. Previous work (Zhang in Cluster variables and perfect matchings of subgraphs of the dP₃ lattice, http://www.math.umn.edu/~reiner/REU/Zhang2012.pdf. arXiv:1511.0655, 2012; Leoni et al. in J Phys A Math Theor 47:474011, 2014), which arose during the second author’s mentoriship of undergraduates, and more recently of both authors (Lai and Musiker in Commun Math Phys 356(3):823–881, 2017), analyzed the cluster algebra associated with the cone over dP₃, the del Pezzo surface of degree 6 (CP² blown up at three points). By investigating sequences of toric mutations, those occurring only at vertices with two incoming and two outgoing arrows, in this cluster algebra, we obtained a family of cluster variables that could be parameterized by Z³ and whose Laurent expansions had elegant combinatorial interpretations in terms of dimer partition functions (in most cases). While the earlier work (Lai and Musiker 2017; Zhang 2012; Leoni et al. 2014) focused exclusively on one possible initial seed for this cluster algebra, there are in total four relevant initial seeds (up to graph isomorphism). In the current work, we explore the combinatorics of the Laurent expansions from these other initial seeds and how this allows us to relate enumerations of perfect matchings on Dungeons to Dragons.