Random unfriendly seating arrangement in a dining table

TL;DR

This paper analyzes the distribution of occupied seats in a two-row unfriendly seating arrangement, deriving an asymptotic probability generating function and establishing a local limit theorem using Riccati equations.

Contribution

It introduces a novel identity for the probability generating function and applies Riccati equations to derive asymptotic results for the seating model.

Findings

Derived an unusual identity for the probability generating function

Established a local limit theorem with optimal convergence rate

Provided asymptotic expansion for the probability generating function

Abstract

A detailed study is made of the number of occupied seats in an unfriendly seating scheme with two rows of seats. An unusual identity is derived for the probability generating function, which is itself an asymptotic expansion. The identity implies particularly a local limit theorem with optimal convergence rate. Our approach relies on the resolution of Riccati equations.