Related papers: A Matrix Model for Higher-Genus Fuss--Catalan Numb…
This paper provides an alternate proof to parts of the Goulden-Slofstra formula for enumerating two vertex maps by genus, which is an extension of the famous Harer-Zagier formula that computes the Euler characteristic of the moduli space of…
We define q-Catalan bases which are a generalization of the q-polynomials z^n(z,q)_n. The determination of their dual bases involves some q-power series termed dual coefficients. We show how these dual coefficients occur in the solution of…
In this note we give explicitly a family of probability densities, the moments of which are Fuss-Catalan numbers. The densities appear naturally in random matrices, free probability and other contexts.
The ECO method and the theory of Catalan-like numbers introduced by Aigner seems two completely unrelated combinatorial settings. In this work we try to establish a bridge between them, aiming at starting a (hopefully) fruitful study on…
We describe arithmetic algorithms on a canonical number representation based on the Catalan family of combinatorial objects specified as a Haskell type class. Our algorithms work on a {\em generic} representation that we illustrate on…
In this note we introduce a determinant and then give its evaluating formula. The determinant turns out to be a generalization of the well-known ballot and Fuss-Catalan numbers, which is believed to be new. The evaluating formula is proved…
We study two classes of permutations intimately related to the visual proof of Spitzer's lemma and Huq's generalization of the Chung-Feller theorem. Both classes of permutations are counted by the Fuss-Catalan numbers. The study of one…
Recently we introduced the hypergraph matrix model (HMM), a Hermitian matrix model generalizing the classical Gaussian Unitary Ensemble (GUE). In this model the Gaussians of the GUE, whose moments count partitions of finite sets into pairs,…
Let $f:\mathbb{Z}\longrightarrow \{ \times \cdot\}$ be a function such that $f(a) = \cdot$ for all except finitely for many $a \in \mathbb{Z}$. We define a set $\flat f$ of non-intersecting arc (or cap) diagrams satisfying certain…
A linear chord diagram canonically determines a fatgraph and hence has an associated genus $g$. We compute the natural generating function ${\bf C}_g(z)=\sum_{n\geq 0} {\bf c}_g(n)z^n$ for the number ${\bf c}_g(n)$ of linear chord diagrams…
In order to make graphical Gaussian models a viable modelling tool when the number of variables outgrows the number of observations, model classes which place equality restrictions on concentrations or partial correlations have previously…
We consider unicellular maps, or polygon gluings, of fixed genus. A few years ago the first author gave a recursive bijection transforming unicellular maps into trees, explaining the presence of Catalan numbers in counting formulas for…
We study graph bootstrap percolation on the Erd\H{o}s-R\'enyi random graph ${\mathcal G}_{n,p}$. For all $r \ge 5$, we locate the sharp $K_r$-percolation threshold $p_c \sim (\gamma n)^{-1/\lambda}$, solving a problem of Balogh, Bollob\'as…
Motivated by our previous work on hypergeometric functions and the parbelos constant, we perform a deeper investigation on the interplay among generalized complete elliptic integrals, Fourier-Legendre (FL) series expansions, and ${}_p F_q$…
We extend the planar Pfaffian formalism for the evaluation of the Ising partition function to lattices of high topological genus g. The 3D Ising model on a cubic lattice, where g is proportional to the number of sites, is discussed in…
The Catalan triangle, as well as a Fuss-Catalan triangle, enter a problem of counting particular tied arc diagrams. This setting allows us to prove some combinatorial properties of these triangles.
We improve the previously best known lower and upper bounds on the number n_g of numerical semigroups of genus g. Starting from a known recursive description of the tree T of numerical semigroups, we analyze some of its properties and use…
We revisit the enumeration problems of random discrete surfaces (RDS) based on solutions of the discrete equations derived from the matrix models. For RDS made of squares, the recursive coefficients of orthogonal polynomials associated with…
A finitization of the Catalan numbers $ C_n $ can be defined as Euler characteristics of an algebraic structure. We conjecture the existence of a $ q $-deformed version of such structure, and provide evidence for the first two non-trivial…
The $s$-point correlation function of a Gaussian Hermitian random matrix theory, with an external source tuned to generate a multi-critical singularity, provides the intersection numbers of the moduli space for the $p$-th spin curves…