Multivariate asymptotic analysis of set partitions: Focus on blocks of fixed size

Using the Saddle point method and multiseries expansions, we obtain from the exponential formula and Cauchy’s integral formula, asymptotic results for the number T(n; m; k) of partitions of n labeled objects with m blocks of fixed size k. We analyze the central and non-central region. In the region m = n=k - n^α; 1 α 1=2, we analyze the dependence of T(n; m; k) on α. This paper fits within the framework of Analytic Combinatorics.