A Combinatorial Problem in Cinema Seating

Abstract

We address a problem concerning cinema audiences: ``A cinema has $n$ seats numbered from $1$ to $n$, and there are $n$ people with tickets numbered from $1$ to $n$. People enter the cinema in order. If someone has the ticket number $i$, they can choose seats whose numbers are multiples of $i$. They should exit the cinema if the permitted seats are occupied by previous audience members. In how many ways can they be seated under these conditions?" We give an algorithm to create the list of situations that meet these conditions. We also focus on finding the number of situations in two special cases: when exactly one seat is unoccupied whose total number is denoted by $ω(n)$, and when all audiences $1, \ldots, n-1$ are seated, whose total number is denoted by $ψ(n)$. Giving the recursive formula $ψ(n)=1+\sum_{d|n, d\neq n}ψ(d)$ with the initial value $ψ(1)=1$, we provide an explicit formula for $ψ(p^αq^β)$, where $p$ and $q$ are distinct prime numbers. Furthermore, we show that $ω(n)=-n+\sum_{i=1}^nψ(i)$.