Scattering diagrams, tight gradings, and generalized positivity

We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets. These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras. To a continued fraction $[a_{1},a_{2},\ldots ,a_{n}]$ we associate a snake graph ${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$ such that the continued fraction is the quotient of the number of perfect matchings of ${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$ and ${\mathcal{G}}[a_{2},\ldots ,a_{n}]$. We also show that snake graphs are in bijection with continued fractions. We then apply this connection between cluster algebras and continued fractions in two directions. First we use results from snake graph calculus to obtain new identities for the continuants of continued fractions. Then we apply the machinery of continued fractions to cluster algebras and obtain explicit direct formulas for quotients of elements of the cluster algebra as continued fractions of Laurent polynomials in the initial variables. Building on this formula, and using classical methods for infinite periodic continued fractions, we also study the asymptotic behavior of quotients of elements of the cluster algebra.

In this paper we study consequences of the results of Kang et al. [ Monoidal categorification of cluster algebras , J. Amer. Math. Soc. 31 (2018), 349–426] on a monoidal categorification of the unipotent quantum coordinate ring $A_{q}(\mathfrak{n}(w))$ together with the Laurent phenomenon of cluster algebras. We show that if a simple module $S$ in the category ${\mathcal{C}}_{w}$ strongly commutes with all the cluster variables in a cluster $[\mathscr{C}]$ , then $[S]$ is a cluster monomial in $[\mathscr{C}]$ . If $S$ strongly commutes with cluster variables except for exactly one cluster variable $[M_{k}]$ , then $[S]$ is either a cluster monomial in $[\mathscr{C}]$ or a cluster monomial in $\unicode[STIX]{x1D707}_{k}([\mathscr{C}])$ . We give a new proof of the fact that the upper global basis is a common triangular basis (in the sense of Qin [ Triangular bases in quantum cluster algebras and monoidal categorification conjectures , Duke Math. 166 (2017), 2337–2442]) of the localization $\widetilde{A}_{q}(\mathfrak{n}(w))$ of $A_{q}(\mathfrak{n}(w))$ at the frozen variables. A characterization on the commutativity of a simple module $S$ with cluster variables in a cluster $[\mathscr{C}]$ is given in terms of the denominator vector of $[S]$ with respect to the cluster $[\mathscr{C}]$ .

On cluster-tilting graphs for hereditary categories

We apply freezing operators to relate different (quantum) upper cluster algebras. We prove that these operators send localized (quantum) cluster monomials to localized (quantum) cluster monomials. They also send bases to bases in many cases. In addition, the bases constructed via freezing coincide with those constructed via localization.

It has been known that several objects such as cluster variables, coefficients, seeds, and $Y$-seeds in different cluster patterns with common exchange matrices share the same periodicity under mutations. We call it synchronicity phenomenon in cluster patterns. In this expository note we explain the mechanism of synchronicity based on several fundamental results on cluster algebra theory such as separation formulas, sign-coherence, Laurent positivity, duality, and detropicalization obtained by several authors. We also show that all synchronicity properties studied in this paper are naturally extended to cluster patterns of generalized cluster algebras, up to the Laurent positivity conjecture.

We generalize the construction of the bangle, band and bracelet bases for cluster algebras from unpunctured orbifolds to the case where there is only one marked point on the boundary.

In connection with generalized cluster algebras we introduce a certain generalization of the celebrated Rogers dilogarithm, which we call the Rogers dilogarithms of higher degree. We show that there is an identity of these generalized Rogers dilogarithms associated with any period of seeds of a generalized cluster algebra.

We introduce the notion of a $Y$ -pattern with coefficients and its geometric counterpart: an $\mathcal {X}$ -cluster variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially completed $\mathcal {X}$ -cluster variety $\widehat {\mathcal {X} }$ to the toric variety associated to its g -fan. Moreover, we show that the fibers of this family are stratified in a natural way, with strata the specially completed $\mathcal {X}$ -varieties encoded by $\operatorname {Star}(\tau )$ for each cone $\tau$ of the $\mathbf {g}$ -fan. These strata degenerate to the associated toric strata of the central fiber. We further show that the family is cluster dual to $\mathcal {A}_{\mathrm {prin}}$ of Gross, Hacking, Keel and Kontsevich [ Canonical bases for cluster algebras , J. Amer. Math. Soc. 31 (2018), 497–608], and the fibers cluster dual to $\mathcal {A} _t$ . Finally, we give two applications. First, we use our construction to identify the toric degeneration of Grassmannians from Rietsch and Williams [ Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians , Duke Math. J. 168 (2019), 3437–3527] with the Gross–Hacking–Keel–Kontsevich degeneration in the case of $\operatorname {Gr}_2(\mathbb {C} ^{5})$ . Next, we use it to link cluster duality to Batyrev–Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry.

Nakajima’s graded quiver varieties naturally appear in the study of bases of cluster algebras. One particular family of these varieties, namely the bipartite determinantal varieties, can be defined for any bipartite quiver and gives a vast generalization of classical determinantal varieties with broad applications to algebra, geometry, combinatorics, and statistics. The ideals that define bipartite determinantal varieties are called bipartite determinantal ideals. We provide an elementary method of proof showing that the natural generators of a bipartite determinantal ideal form a Gröbner basis, using an S-polynomial construction method that relies on the Leibniz formula for determinants. This method is developed from an idea by Narasimhan and Caniglia–Guccione–Guccione. As applications, we study the connection between double determinantal ideals (which are bipartite determinantal ideals of a quiver with two vertices) and tensors, and provide an interpretation of these ideals within the context of algebraic statistics.

The doubloon polynomials are generating functions for a class of combinatorial objects called normalized doubloons by the compressed major index. They provide a refinement of the q-tangent numbers and also involve two major specializations: the Poupard triangle and the Catalan triangle.

ECO is a method for the recursive generation, and thereby also the enumeration of classes of combinatorial objects. It has already found successful application in recent literature both to the exhaustive generation and to the uniform random generation of various objects classified according to several parameters of interest, as well as to their enumeration. In this paper we extend this approach to the generation and enumeration of some classes of convex polyominoes. We begin with a review of the ECO method and of the closely related notion of a succession rule. From this background, we develop the following principal findings: i) ECO constructions for both column-convex and convex polyominoes; ii) translations of these constructions into succession rules; iii) the consequent deduction of the generating functions of column-convex and of convex polyominoes according to their semi-perimeter, first of all analytically by means of the so-called kernel method, and then in a more novel manner by drawing on some ideas of Fedou and Garcia; iv) algorithms for the exhaustive generation of column convex and of convex polyominoes which are based on the ECO constructions of these object and which are shown to run in constant amortized time.

A methodology for plane tree enumeration

Letnandkbe integers, withand. Ansemi‐Latin squareSis anarray, whose entries arek‐subsets of an‐set, the set ofsymbolsofS, such that each symbol ofSis in exactly one entry in each row and exactly one entry in each column ofS. Semi‐Latin squares form an interesting class of combinatorial objects which are useful in the design of comparative experiments. We say that ansemi‐Latin squareSisuniformif there is a constant μ such that any two entries ofS, not in the same row or column, intersect in exactly μ symbols (in which case). We prove that a uniformsemi‐Latin square is Schur‐optimal in the class ofsemi‐Latin squares, and so is optimal (for use as an experimental design) with respect to a very wide range of statistical optimality criteria. We give a simple construction to make ansemi‐Latin squareSfrom a transitive permutation groupGof degreenand order, and show how certain properties ofScan be determined from permutation group properties ofG. IfGis 2‐transitive thenSis uniform, and this provides us with Schur‐optimal semi‐Latin squares for many values ofnandkfor which optimalsemi‐Latin squares were previously unknown for any optimality criterion. The existence of a uniformsemi‐Latin square for all integersis shown to be equivalent to the existence ofmutually orthogonal Latin squares (MOLS) of ordern. Although there are not even two MOLS of order 6, we construct uniform, and hence Schur‐optimal,semi‐Latin squares for all integers. & 2012 Wiley Periodicals, Inc. J. Combin. Designs 00: 1–13, 2012

de Bruijn cycles for neural decoding

In this paper we use ECO method to enumerate restricted classes of combinatorial objects; if a succession rule Ω describes the construction of a class, then the restricted class can be described by means of an approximating succession rule Ω k obtained from Ω; we determine finite approximating rules for various classes of paths, and the approximation of the corresponding algebraic language with a regular one.