Stephen Melczer

In the quarter plane, five lattice path models with unit steps have resisted the otherwise general approach featured in recent works by Fayolle, Kurkova and Raschel. Expand

We consider the number of partitions of $n$ whose Young diagrams fit inside an $m \times \ell$ rectangle; equivalently, we study the coefficients of the $q$-binomial coefficient $\binom{m+\ell}{m}_q$. Expand

We consider d-dimensional lattice path models restricted to the first orthant whose defining step sets exhibit reflective symmetry across every axis. Expand

The field of analytic combinatorics in several variables (ACSV) has shown how to deriveasymptotics for the coefficients of certain D-finite functions represented by diagonals ofmultivariate rational functions. Expand

We consider the enumeration of walks on the non-negative lattice $\mathbb{N}^d$, with steps defined by a set $\mathcal{S} \subset \{-1, 0, 1\}^D \setminus \{\mathbf{0}\}$. Expand

We explain a discontinuous drop in the exponential growth rate for certain multivariate generating functions at a critical parameter value, in even dimensions $d \geq 4$. Expand

We prove solution uniqueness for the genus one Canham variational problem arising in the shape prediction of biomembranes. Expand