In previous work [LM17], Tri Lai and the second author studied a family of subgraphs of the dP3 brane tiling, called Aztec castles, whose dimer partition functions provide combinatorial formulas for cluster variables resulting from mutations of the quiver associated with the del Pezzo surface dP3. In our paper, we investigate a variant of the dP3 quiver by considering a second alphabet of variables that breaks the symmetries of the relevant recurrences. This deformation is motivated by the theory of cluster algebras with principal coefficients introduced by Fomin and Zelevinsky. Our main result gives an explicit formula extending previously known generating functions for dP3 cluster variables by using Aztec castles and constructing their associated minimal matchings.

We study a twisted version of Fraser, Lam, and Le's higher boundary measurement map, using face weights instead of edge weights, thereby providing Laurent polynomial expansions, in Pl & uuml;cker coordinates, for twisted web immanants for Grassmannians. In some small cases, Fraser, Lam, and Le observe a phenomenon they call "web duality", where web immanants coincide with web invariants, and they conjecture that this duality corresponds to transposing the standard Young tableaux that index basis webs. We show that this duality continues to hold for a large set of SL3 and SL4 webs. Combining this with our twisted higher boundary measurement map, we recover and extend formulas of Elkin-Musiker-Wright for twists of certain cluster variables. We also provide evidence supporting conjectures of Fomin-Pylyavskyy as well as one by Cheung-Dechant-He-Heyes-Hirst-Li concerning classification of cluster variables of low Pl & uuml;cker degree in C [Gr(3, n)].

This paper studies a lattice structure for almost perfect matchings on certain planar, bipartite (plabic) graphs embedded in a disk. Postnikov's boundary measurement map, and subsequent related work, yielded that plabic graphs parameterize positroid cells within the totally nonnegative Grassmannian with the map itself given in terms of almost perfect matchings with fixed boundary condition. For finite planar bipartite graphs, Propp introduced a distributive lattice structure on their set of perfect matchings. Subsequently Muller–Speyer, provided this distributive lattice structure on the aforementioned almost perfect matchings with fixed boundary condition. Their work also identified the extremal matchings of this lattice for boundary conditions that coincide with face labels of the plabic graph given by the positroid structure. We extend this by giving an explicit construction of extremal matchings in terms of height functions and show that all possible boundary conditions of an almost perfect matching can be obtained within this construction.

The set of perfect matchings of a connected bipartite plane graph G has the structure of a distributive lattice, as shown by Propp, where the partial order is induced by the height of a matching. In this article, our focus is the dimer face polynomial of G, which is the height generating function of all perfect matchings of G. We connect the dimer face polynomial on the one hand to knot theory, and on the other to cluster algebras. We show that certain dimer face polynomials specialize to Alexander polynomials of links, highlighting another combinatorial view of the Alexander polynomial. We also show that an arbitrary dimer face polynomial is an F-polynomial in the cluster algebra whose initial quiver is dual to the graph G. As a result, we recover a recent representation theoretic result of Bazier-Matte and Schiffler that connects F-polynomials and Alexander polynomials, albeit from a very different, dimer-based perspective. As another application of our results, we also show that all nonvanishing Plücker coordinates on open positroid varieties are cluster monomials.

Snake graphs are a class of planar graphs that are important in the theory of cluster algebras. Indeed, the Laurent expansions of the cluster variables in cluster algebras from surfaces are given as weight generating functions for 1-dimer covers (or perfect matchings) of snake graphs. Moreover, the enumeration of 1-dimer covers of snake graphs provides a combinatorial interpretation of continued fractions. In particular, the number of 1-dimer covers of the snake graph $\mathscr{G}[a_1,\dots,a_n]$ is the numerator of the continued fraction $[a_1,\dots,a_n]$. This number is equal to the top left entry of the matrix product $\left(\begin{smallmatrix} a_1&1\\1&0 \end{smallmatrix}\right) \cdots \left(\begin{smallmatrix} a_n&1\\1&0 \end{smallmatrix}\right)$. In this paper, we give enumerative results on $m$-dimer covers of snake graphs. We show that the number of $m$-dimer covers of the snake graph $\mathscr{G}[a_1,\ldots,a_n]$ is the top left entry of a product of analogous $(m+1)$-by-$(m+1)$ matrices. We discuss how our enumerative results are related to other known combinatorial formulas, and we suggest a generalization of continued fractions based on our methods. These generalized continued fractions provide some interesting open questions and a possibly novel approach towards Hermite's problem for cubic irrationals.

We study rank-2 cluster scattering diagrams through moduli spaces of quiver representations and a recently developed combinatorial framework of tight gradings. Combining quiver-theoretic and combinatorial methods, we prove and extend a collection of conjectures posed by Elgin--Reading--Stella concerning the structural and enumerative properties of the wall-function coefficients. The tight grading perspective also provides a new proof of the Weyl group symmetry of the scattering diagram.

We provide a superalgebraic analogue of Markov numbers, which are defined as the Grassmann integer solutions to the equation x^2 + y^2 + z^2 + (xy + yz + xz)ϵ = 3(1 + ϵ)xyz, as well as applications to the Decorated Super Teichmüller spaces associated to the once-punctured torus and certain annuli. We conclude with further directions for study.

We discuss a product formula for $F$-polynomials in cluster algebras, and provide two proofs. One proof is inductive and uses only the mutation rule for $F$-polynomials. The other is based on the Fock-Goncharov decomposition of mutations. We conclude by expanding this product formula as a sum and illustrate applications. This expansion provides an explicit combinatorial computation of $F$-polynomials in a given seed that depends only on the $\mathbf{c}$-vectors and $\mathbf{g}$-vectors along a finite sequence of mutations from the initial seed to the given seed.

This is a sequel to the second and third author's Mixed Dimer Configuration Model in Type D Cluster Algebras where we extend our model to work for quivers that contain oriented cycles. Namely, we extend a combinatorial model for F-polynomials for type D_n using dimer and double dimer configurations. In particular, we give a graph theoretic recipe that describes which monomials appear in such F-polynomials, as well as a graph theoretic way to determine the coefficients of each of these monomials. To prove this formula, we provide an explicit bijection between mixed dimer configurations and dimension vectors of submodules of an indecomposable Jacobian algebra module.

We give a combinatorial interpretation for certain cluster variables in Grassmannian cluster algebras in terms of double and triple dimer configurations. More specifically, we examine several Gr(3,n) cluster variables that may be written as degree two or degree three polynomials in terms of Pl\"ucker coordinates, and give generating functions for their images under the twist map - a cluster algebra automorphism introduced in work of Berenstein-Fomin-Zelevinsky. The generating functions range over certain double or triple dimer configurations on an associated plabic graph, which we describe using particular non-crossing matchings or webs (as defined by Kuperberg), respectively. These connections shed light on a recent conjecture of Cheung et al., extend the concept of web duality introduced in a paper of Fomin-Lam-Le, and more broadly make headway on understanding Grassmannian cluster algebras for Gr(3,n).

For an arc on a bordered surface with marked points, we associate a holonomy matrix using a product of elements of the supergroup OSp(1|2), which defines a flat OSp(1|2)-connection on the surface. We show that our matrix formula of an arc yields its super λ-length in Penner-Zeitlin's decorated super Teichmüller space. This generalizes the matrix formula of Fock-Goncharov and Musiker-Williams. We also prove that our matrix formulas agree with the combinatorial formulas given in the authors' previous works. As an application, we use our matrix formula in the case of an annulus to obtain new results on super Fibonacci numbers.

We define a combinatorial model for F-polynomials and g-vectors for type Dn cluster algebras where the associated quiver is acyclic. Our model utilizes a com-bination of dimer configurations and double dimer configurations which we refer to as mixed dimer configurations. In particular, we give a graph theoretic recipe that describes which monomials appear in such F-polynomials, as well as a graph theo-retic way to determine the coefficients of each of these monomials. In addition, we prove that a weighting on our mixed dimer configuration model yields the associated g-vector. To prove this formula, we use a combinatorial formula due to Thao Tran (arXiv:0911.4462, 2009) and provide explicit bijections between her combinatorial model and our own.

We study labeled chip-firing on binary trees starting with 2^n-1 chips initially placed at the root. We prove a sorting property of terminal configurations of the process. We also analyze the end game moves poset and prove that this poset is a modular lattice.

We give a construction of generalized cluster varieties and generalized cluster scattering diagrams for reciprocal generalized cluster algebras, the latter of which were defined by Chekhov and Shapiro. These constructions are analogous to the structures given for ordinary cluster algebras in the work of Gross, Hacking, Keel, and Kontsevich. As a consequence of these constructions, we are also able to construct theta functions for generalized cluster algebras, again in the reciprocal case, and demonstrate a number of their structural properties.

In a recent paper, the authors gave combinatorial formulas for the Laurent expansions of super λ-lengths in a marked disk, generalizing Schiffler’s T -path formula. In the present paper, we give an alternate combinatorial expression for these super λ-lengths in terms of double dimer covers on snake graphs. This generalizes the dimer formulas of Musiker, Schiffler, and Williams.

We initiate the study of applications of machine learning to Seiberg duality, focusing on the case of quiver gauge theories, a problem also of interest in mathematics in the context of cluster algebras. Within the general theme of Seiberg duality, we define and explore a variety of interesting questions, broadly divided into the binary determination of whether a pair of theories picked from a series of duality classes are dual to each other, as well as the multiclass determination of the duality class to which a given theory belongs. We study how the performance of machine learning depends on several variables, including number of classes and mutation type (finite or infinite). In addition, we evaluate the relative advantages of Naive Bayes classifiers versus convolutional neural networks. Finally, we also investigate how the results are affected by the inclusion of additional data, such as ranks of gauge/flavor groups and certain variables motivated by the existence of underlying Diophantine equations. In all questions considered, high accuracy and confidence can be achieved.

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 $$dP_{3}$$ 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 $$\mathbf {dP_3}$$, the del Pezzo surface of degree 6 ($${\mathbb {C}}{\mathbb {P}}^2$$ 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 (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.

Birational rowmotion is an action on the space of assignments of rational functions to the elements of a finite partially-ordered set (poset). It is lifted from the well-studied rowmotion map on order ideals (equivariantly on antichains) of a poset $P$, which when iterated on special posets, has unexpectedly nice properties in terms of periodicity, cyclic sieving, and homomesy (statistics whose averages over each orbit are constant) [AST11, BW74, CF95, Pan09, PR13, RuSh12,RuWa15+,SW12, ThWi17, Yil17. In this context, rowmotion appears to be related to Auslander-Reiten translation on certain quivers, and birational rowmotion to $Y$-systems of type $A_m \times A_n$ described in Zamolodchikov periodicity. We give a formula in terms of families of non-intersecting lattice paths for iterated actions of the birational rowmotion map on a product of two chains. This allows us to give a much simpler direct proof of the key fact that the period of this map on a product of chains of lengths $r$ and $s$ is $r+s+2$ (first proved by D.~Grinberg and the second author), as well as the first proof of the birational analogue of homomesy along files for such posets.

Originally studied by Conway and Coxeter, friezes appeared in various recreational mathematics publications in the 1970s. More recently, in 2015, Baur, Parsons, and Tschabold constructed periodic infinite friezes and related them to matching numbers in the once-punctured disk and annulus. In this paper, we study such infinite friezes with an eye towards cluster algebras of type D and affine A, respectively. By examining infinite friezes with Laurent polynomial entries, we discover new symmetries and formulas relating the entries of this frieze to one another. Lastly, we also present a correspondence between Broline, Crowe and Isaacs’s classical matching tuples and combinatorial interpretations of elements of cluster algebras from surfaces.