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The operation of twinning a graph at a vertex was introduced by Foley, Ho\`ang, and Merkel (2019), who conjectured that twinning preserves $e$-positivity of the chromatic symmetric function. A counterexample to this conjecture was given by Li, Li, Wang, and Yang (2021). In this paper, we prove that $e$-positivity is preserved by the twinning operation on cycles, by giving an $e$-positive generating function for the chromatic symmetric function, as well as an $e$-positive recurrence. We derive similar $e$-positive generating functions and recurrences for twins of paths. Our methods make use of the important triple deletion formulas of Orellana and Scott (2014), as well as new symmetric function identities., Comment: 35 pages, 2 tables, 9 figures

We construct a bijection between certain Deodhar components of a braid variety constructed from an affine Kac-Moody group of type $A_{n-1}$ and vertex-labeled trees on $n$ vertices. By an argument of Galashin, Lam, and Williams using Opdam's trace formula in the affine Hecke algebra and an identity due to Haglund, we obtain an elaborate new proof for the enumeration of the number of vertex-labeled trees on $n$ vertices., Comment: 21 pages

We consider certain generalizations of gentle algebras that we call semilinear locally gentle algebras. These rings are examples of semilinear clannish algebras as introduced by the second author and Crawley-Boevey. We generalise the notion of a nodal algebra from work of Burban and Drozd and prove that semilinear gentle algebras are nodal by adapting a theorem of Zembyk. We also provide a geometric realization of Zembyk's proof, which involves cutting the surface into simpler pieces in order to endow our locally gentle algebra with a semilinear structure. We then consider this surface glued back together, with the seams in place, and use it to give a geometric model for the finite-dimensional modules over the semilinear locally gentle algebra., Comment: 36 pages, comments welcome

Graph LP algebras are a generalization of cluster algebras introduced by Lam and Pylyavskyy. We provide a combinatorial proof of positivity for certain cluster variables in these algebras. This proof uses a hypergraph generalization of snake graphs, a class of planar graphs which were used by Musiker, Schiffler, and Williams to prove positivity for cluster algebras from surfaces. These results extend those given in our previous paper, where we used a related combinatorial object known as a $T$-path.

Generalized cluster algebras from orbifolds were defined by Chekhov and Shapiro to give a combinatorial description of their Teichm\"uller spaces. One can also assign a gentle algebra to a triangulated orbifold, as in the work of Labardini-Fragoso and Mou. In this work, we show that the Caldero-Chapoton map and the snake graph expansion map agree for arcs in triangulated orbifolds and arc modules, and similarly for closed curves and certain band modules. As a consequence, we have a bijection between some indecomposable modules over a gentle algebra and cluster variables in a generalized cluster algebra where both algebras arise from the same triangulated orbifold., Comment: Comments welcome

A frieze on a polygon is a map from the diagonals of the polygon to an integral domain which respects the Ptolemy relation. Conway and Coxeter previously studied positive friezes over $\mathbb{Z}$ and showed that they are in bijection with triangulations of a polygon. We extend their work by studying friezes over $\mathbb Z[\sqrt{2}]$ and their relationships to dissections of polygons. We largely focus on the characterization of unitary friezes that arise from dissecting a polygon into triangles and quadrilaterals. We identify a family of dissections that give rise to unitary friezes and conjecture that this gives a complete classification of dissections which admit a unitary frieze.

We explore a generalization of the Markov numbers that is motivated by a specific generalized cluster algebra arising from an orbifold, in the sense of Chekhov and Shapiro. We give an explicit algorithm for computing these generalized Markov numbers and exhibit several patterns analogous to those that appear within the ordinary Markov numbers. Along the way, we present formulas related to continued fractions and snake graphs., Comment: 33 pages, many figures, comments welcome

LP algebras, introduced by Lam and Pylyavskyy, are a generalization of cluster algebras. These algebras are known to have the Laurent phenomenon, but positivity remains conjectural. Graph LP algebras are finite LP algebras encoded by a graph. For the graph LP algebra defined by a tree, we define a family of clusters called rooted clusters. We prove positivity for these clusters by giving explicit formulas for each cluster variable. We also give a combinatorial interpretation for these expansions using a generalization of $T$-paths.

Finite frieze patterns with entries in $\mathbb{Z}[\lambda_{p_1},\ldots,\lambda_{p_s}]$ where $\{p_1,\ldots,p_s\} \subseteq \mathbb{Z}_{\geq 3}$ and $\lambda_p = 2 \cos(\pi/p)$ were shown to have a connection to dissected polygons by Holm and Jorgensen. We extend their work by studying the connection between infinite frieze patterns with such entries and dissections of annuli and once-punctured discs. We give an algorithm to determine whether a frieze pattern with entries in $\mathbb{Z}[\lambda_{p_1},\ldots,\lambda_{p_s}]$, finite or infinite, comes from a dissected surface. We introduce quotient dissections as a realization for some frieze patterns unrealizable by an ordinary dissection. We also introduce two combinatorial interpretations for entries of frieze patterns from dissected surfaces. These interpretations are a generalization of matchings introduced by Broline, Crowe, and Isaacs for finite frieze patterns over $\mathbb{Z}$., Comment: 52 pages, many figures, comments welcome

We give an explicit combinatorial formula for the Laurent expansion of any arc or closed curve on an unpunctured triangulated orbifold. We do this by extending the snake graph construction of Musiker, Schiffler, and Williams to unpunctured orbifolds. In the case of an ordinary arc, this gives a combinatorial proof of positivity to the generalized cluster algebra from this orbifold.

Juggling patterns can be described by a closed walk in a (directed) state graph, where each vertex (or state) is a landing pattern for the balls and directed edges connect states that can occur consecutively. The number of such patterns of length $n$ is well known, but a long-standing problem is to count the number of prime juggling patterns (those juggling patterns corresponding to cycles in the state graph). For the case of $b=2$ balls we give an expression for the number of prime juggling patterns of length $n$ by establishing a connection with partitions of $n$ into distinct parts. From this we show the number of two-ball prime juggling patterns of length $n$ is $(\gamma-o(1))2^n$ where $\gamma=1.32963879259...$. For larger $b$ we show there are at least $b^{n-1}$ prime cycles of length $n$.

We consider a generalization of Eulerian numbers which count the number of placements of $cn$ "rooks" on an $n\times n$ board where there are exactly $c$ rooks in each row and each column, and exactly $k$ rooks below the main diagonal. The standard Eulerian numbers correspond to the case $c=1$. We show that for any $c$ the resulting numbers are symmetric and give generating functions of these numbers for small values of $k$., Comment: 15 pages