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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 $m=n/k-n^\al,\quad 1>\al>1/2$, we analyze the dependence of $T(n,m,k)$ on $\al$. This paper fits within the framework of Analytic Combinatorics.

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