  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 [Zha, LMNT14], which
arose during the second author's monitorship of undergraduates, and more
recently of both authors [LM17], analyzed the cluster algebra associated to the
cone over $\mathbf{dP_3}$, the del Pezzo surface of degree $6$ ($\mathbb{CP}^2$
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 $\mathbb{Z}^3$ and whose Laurent expansions had elegant
combinatorial interpretations in terms of dimer partition functions (in most
cases). While the earlier work [Zha, LMNT14, LM17] 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.
