Related papers: Caustics, counting maps and semi-classical asympto…

We establish the asymptotic expansion in $\beta$ matrix models with a confining, off-critical potential, in the regime where the support of the equilibrium measure is a union of segments. We first address the case where the filling…

This work has two main purposes. On the one side we investigate in this work a question of H. Esnault on congruence formula in a construction of H. Esnault and C. Xu for the number of rational points on the closed fiber of a singular model…

Recently, a new class of words, denoted by L_n, was shown to be in bijection with a subset of the Dyck paths of length 2n having cardinality given by the (n-1)-st Catalan number. Here, we consider statistics on L_n recording the number of…

A preferential arrangement of a set $X_n=\{1,2,...,n\}$ is an ordered partition of the set $X_n$ induced with a linear order. Separation of blocks of a preferential arrangement with bars result in the notation of barred preferential…

This paper examines the coefficient problems for the class of semigroup generators, a topic in complex dynamics that has recently been studied in context of geometric function theory. Further, sharp bounds of coefficient functional such as…

We provide new formulas for the coefficients in the partial fraction decomposition of the restricted partition generating function. These techniques allow us to partially resolve a recent conjecture of Sills and Zeilberger. We also describe…

We consider a number of combinatorial problems in which rational generating functions may be obtained, whose denominators have factors with certain singularities. Specifically, there exist points near which one of the factors is asymptotic…

A conformal partition function ${\cal P}_n^m(s)$, which arose in the theory of Diophantine equations supplemented with additional restrictions, is concerned with {\it self-dual symmetric polynomials} -- reciprocal ${\sf R}^{\{m\}}_ {S_n}$…

We study generating functions in the context of Rota-Baxter algebras. We show that exponential generating functions can be naturally viewed in a very special case of complete free commutative Rota-Baxter algebras. This allows us to use free…

In a recent paper, Bacher and de la Harpe study the conjugacy growth series of finitary permutation groups. In the course of studying the coefficients of a series related to the finitary alternating group, they introduce generalized…

The exponential generating functions of {n^(n+m)} for arbitrary integer m are expressed as rational functions of the e.g.f. of {n^(n-1)} [the tree function] and then of the e.g.f. of {n^n} [the endofunction function]. The coefficients in…

Following our recent letter, we study in detail an entry-wise diffusion of non-hermitian complex matrices. We obtain an exact partial differential equation (valid for any matrix size $N$ and arbitrary initial conditions) for evolution of…

Filling functions are asymptotic invariants of finitely presentable groups; the seminal work on the subject is by M.Gromov. They record features of combinatorial homotopy discs (van Kampen diagrams) filling loops in Cayley 2-complexes.…

We study compositions of a positive integer $n$ in which the occurrence of even parts larger than a fixed threshold $k$ is controlled. More precisely, for each composition $m=(m_1,\dots,m_r)$ we consider the number of even parts strictly…

We derive asymptotic formulae for the coefficients of bivariate generating functions with algebraic and logarithmic factors. Logarithms appear when encoding cycles of combinatorial objects, and also implicitly when objects can be broken…

Inspired by Gansner's elegant $k$-trace generating function for rectangular plane partitions, we introduce two novel operators, $\varphi_{z}$ and $\psi_{z}$, along with their combinatorial interpretations. Through these operators, we derive…

We introduce a class of stochastic processes with reinforcement consisting of a sequence of random partitions $\{\mathcal{P}_t\}_{t \ge 1}$, where $\mathcal{P}_t$ is a partition of $\{1,2,\dots, Rt\}$. At each time~$t$,~$R$ numbers are…

We explain a discontinuous drop in the exponential growth rate for certain multivariate generating functions at a critical parameter value, in even dimensions d at least 4. This result depends on computations in the homology of the…

There exist a number of well known multiplicative generating functions for series of Schur functions. Amongst these are some related to the dual Cauchy identity whose expansion coefficients are rather simple, and in some cases periodic in…

We present a strategy for computing asymptotics of coefficients of $d$-variate algebraic generating functions. Using known constructions, we embed the coefficient array into an array represented by a rational generating functions in $d+1$…