Pascal's formulas and vector fields

TL;DR

This paper explores how Pascal's formulas generate vector fields from combinatorial triangles and demonstrates their connection to the asymptotic behavior of certain Markov chains.

Contribution

It introduces a novel geometric perspective on combinatorial triangles via vector fields and proves their asymptotic relation to Markov chain sample paths in three cases.

Findings

Vector fields derived from Pascal's formulas match Markov chain limits.

Asymptotic behavior of three combinatorial triangles is rigorously proven.

Field lines correspond to the conjectured limits of sample paths.

Abstract

We consider four examples of combinatorial triangles $\left(T(n,k)\right)_{0\le k\le n}$ (Pascal, Stirling of both types, Euler) : through saddle-point asymptotics, their \emph{Pascal's formulas} define four vector fields, together with their field lines that turn out to be the conjectured limit of sample paths of four well known Markov chains. We prove this asymptotic behaviour in three of the four cases.