Related papers: Lattice Path Enumeration and Its Applications in R…
We study a class of observables in four-dimensional superconformal Yang--Mills theories which, in the planar limit at finite 't Hooft coupling, can be expressed as determinants of semi-infinite matrices built from Bessel functions. This…
In this paper we investigate the number of integer points lying in dilations of lattice path matroid polytopes. We give a characterization of such points as polygonal paths in the diagram of the lattice path matroid. Furthermore, we prove…
We show that the number of linear spaces on a set of $n$ points and the number of rank-3 matroids on a ground set of size $n$ are both of the form $(cn+o(n))^{n^2/6}$, where $c=e^{\sqrt 3/2-3}(1+\sqrt 3)/2$. This is the final piece of the…
Using techniques from the theories of convex polytopes, lattice paths, and indirect influences on directed manifolds, we construct continuous analogues for the binomial coefficients and the Catalan numbers. Our approach for constructing…
We give a new combinatorial proof for the number of convex polyominoes whose minimum enclosing rectangle has given dimensions. We also count the subclass of these polyominoes that contain the lower left corner of the enclosing rectangle…
We transpose Jones' technology and the authors' C*-algebraic techniques to study representations of the Leavitt path algebra L (over an arbitrary row-finite graph) by using its quiver algebra A. We establish an equivalence of categories…
Explicit algebraic area enumeration formulae are derived for various lattice walks generalizing the canonical square lattice walk, and in particular for the triangular lattice chiral walk recently introduced by the authors. A key element in…
Consider an $m\times n$ table $T$ and latices paths $\nu_1,\ldots,\nu_k$ in $T$ such that each step $\nu_{i+1}-\nu_i=(1,1)$, $(1,0)$ or $(1,-1)$. The number of paths from the $(1,i)$-blank (resp. first column) to the $(s,t)$-blank is…
A {\em k-generalized Dyck path} of length $n$ is a lattice path from $(0,0)$ to $(n,0)$ in the plane integer lattice $\mathbb{Z}\times\mathbb{Z}$ consisting of horizontal-steps $(k, 0)$ for a given integer $k\geq 0$, up-steps $(1,1)$, and…
We characterize the quotients among lattice path matroids (LPMs) in terms of their diagrams. This characterization allows us to show that ordering LPMs by quotients yields a graded poset, whose rank polynomial has the Narayana numbers as…
The collection of reflecting hyperplanes of a finite Coxeter group is called a reflection arrangement and it appears in many subareas of combinatorics and representation theory. We focus on the problem of counting regions of reflection…
We study enumerative questions on the moduli space $\mathcal{M}(L)$ of hyperplane arrangements with a given intersection lattice $L$. Mn\"ev's universality theorem suggests that these moduli spaces can be arbitrarily complicated; indeed it…
We discuss several extension properties of matroids and polymatroids and their application as necessary conditions for the existence of different matroid representations, namely linear, folded linear, algebraic, and entropic…
This note presents a formula for the enumerative invariants of arbitrary genus in toric surfaces. The formula computes the number of curves of a given genus through a collection of generic points in the surface. The answer is given in terms…
We present refined enumeration formulas for lattice paths in $\mathbb{Z}^2$ with two kinds of steps, by keeping track of the number of descents (i.e., turns in a given direction), the major index (i.e., the sum of the positions of the…
Let \Gamma=<\alpha, \beta > be a numerical semigroup. In this article we consider several relations between the so-called \Gamma-semimodules and lattice paths from (0,\alpha) to (\beta,0): we investigate isomorphism classes of…
Assuming standard conjectures, we show that the canonical symmetrizing trace evaluated at powers of a Coxeter element produces rational Catalan numbers for irreducible spetsial complex reflection groups. This extends a technique used by…
Noncrossing set partitions, nonnesting set partitions, Dyck paths, and rooted plane trees are four classes of Catalan objects which carry a notion of type. There exists a product formula which enumerates these objects according to type. We…
Following the work of Cano and Diaz, we consider a continuous analog of lattice path enumeration. This allows us to define a continuous version of any discrete object that counts certain types of lattice paths. We define continuous versions…
A Catalan word is one on the alphabet of positive integers starting with $1$ in which each subsequent letter is at most one more than its predecessor. Let $\mathcal{C}_n$ denote the set of Catalan words of length $n$. In this paper, we give…