A continuous analogue of lattice path enumeration

TL;DR

This paper introduces a continuous framework for lattice path enumeration, defining continuous analogs of binomials and multinomials, and demonstrating how to recover discrete counts from these continuous models.

Contribution

It presents a novel continuous analog of lattice path enumeration, including continuous binomials and multinomials, with identities and PDEs, bridging discrete and continuous combinatorial analysis.

Findings

Continuous binomials and multinomials are defined.

Identities and PDEs for continuous analogs are established.

Method to recover discrete counts from continuous models is demonstrated.

Abstract

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 of binomials and multinomials, and describe some identities and partial differential equations they satisfy. Finally, we illustrate a general process to recover discrete combinatorial quantities from their continuous analogs.