Number of singletons in involutions of large size: a central range and large deviation analysis

Abstract

In this paper, we analyze the asymptotic number $I(m,n)$ of involutions of large size $n$ with $m$ singletons. We consider a central region and a non-central region. In the range $m=n–n^{\alpha }$, $0<\alpha <1$, we analyze the dependence of $I(m,n)$ on $\alpha$. This paper fits within the framework of Analytic Combinatorics.