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

Year 2017, Volume: 4 Issue: 1, 75 - 91, 11.01.2017

Guy Louchard

https://doi.org/10.13069/jacodesmath.37019

Abstract

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.

Keywords

Set partitions, Bell numbers, Asymptotics, Saddle point method, Multiseries expansions, Analytic combinatorics

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