Related papers: Gale-Robinson Quivers and Principal Coefficients

We introduce a family of sequence transformations, defined via partial Bell polynomials, that may be used for a systematic study of a wide variety of problems in enumerative combinatorics. This family includes some of the transformations…

Connected the generalized Goncharov polynomials associated to a pair ($\partial,\mathcal{Z}$) if a delta operator $\partial$ and an interpolation grid $\mathcal{Z}$, introduced by Lorentz, Tringali and Yan in [7], with the theory of…

Many combinatorial sequences (for example, the Catalan and Motzkin numbers) may be expressed as the constant term of $P(x)^k Q(x)$, for some Laurent polynomials $P(x)$ and $Q(x)$ in the variable $x$ with integer coefficients. Denoting such…

We define the cluster algebra associated with the Q-system for the Kirillov-Reshetikhin characters of the quantum affine algebra $U_q(\hat{\g})$ for any simple Lie algebra g, generalizing the simply-laced case treated in [Kedem 2007]. We…

We prove the combinatorial invariance of the coefficient of $q$ in Kazhdan--Lusztig polynomials for arbitrary Coxeter groups. As a result, we obtain the Combinatorial Invariance Conjecture, of Lusztig and of Dyer, also for Bruhat intervals…

Bliem and Kousidis (arXiv:1109.4624) recently considered a family of random variables whose distributions are given by the generalized Galois numbers (after normalization). We give probabilistic interpretations of these random variables,…

We consider a class of map, recently derived in the context of cluster mutation. In this paper we start with a brief review of the quiver context, but then move onto a discussion of a related Poisson bracket, along with the Poisson algebra…

We recast homogeneous linear recurrence sequences with fixed coefficients in terms of partial Bell polynomials, and use their properties to obtain various combinatorial identities and multifold convolution formulas. Our approach relies on a…

This paper is devoted to the study of the log-convexity of combinatorial sequences. We show that the log-convexity is preserved under componentwise sum, under binomial convolution, and by the linear transformations given by the matrices of…

We study the $c$-vectors, $g$-vectors, and $F$-polynomials for generalized cluster algebras satisfying a normalization condition and a power condition recovering classical recursions and separation of additions formulas. We establish a…

Three $q$-versions of Lommel polynomials are studied. Included are explicit representations, recurrences, continued fractions, and connections to associated Askey--Wilson polynomials. Combinatorial results are emphasized, including a…

In this article, we prove some factorization results for several classes of polynomials having integer coefficients, which in particular yield several classes of irreducible polynomials. Such classes of polynomials are devised by imposing…

In this paper we obtain some sophisticated combinatorial congruences involving binomial coefficients and confirm two conjectures of the author and Davis. They are closely related to our investigation of the periodicity of the sequence…

Generalizing the famous Bernstein-Kushnirenko Theorem, Khovanskii proved in 1978 a combinatorial formula for the arithmetic genus of the compactification of a generic complete intersection associated to a family of lattice polytopes.…

The (usual) Caldero-Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps "reachable" indecomposable objects to the corresponding cluster…

We introduce a new family of symmetric polynomials $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_{\lambda}$ arising from exactly solvable lattice models associated with the quantised loop algebra $\mathcal{U}_{q}(\mathfrak{sl}_{2}[z^\pm])$. The…

Following our earlier work, where doubly indexed and irreducible over Q two-variable Laguerre polynomials were introduced, we prove for such polynomials some recurrence formulas and obtain a generating function. In addition, we show how…

In this note we augment the poly-Bernoulli family with two new combinatorial objects. We derive formulas for the relatives of the poly-Bernoulli numbers using the appropriate variations of combinatorial interpretations. Our goal is to show…

We investigate generalizations of the Charlier polynomials on the lattice $\mathbb{N}$, on the shifted lattice $\mathbb{N}+1-\beta$ and on the bi-lattice $\mathbb{N}\cup (\mathbb{N}+1-\beta)$. We show that the coefficients of the three-term…

Buch and Fulton conjectured the nonnegativity of the quiver coefficients appearing in their formula for a quiver variety. Knutson, Miller and Shimozono proved this conjecture as an immediate consequence of their ``component formula''. We…