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

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.