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.
Highlights
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
Set partitions parameters have long been a topic of investigation
Set partitions continue to be of interest recently; see Chern et al [1, 2]
Summary
Set partitions parameters have long been a topic of investigation. See, for example, Graham et al [7], Knuth [9], Mansour [13], Stanley [15], for other investigations of set partitions. Let Π(n) be the set of partitions of n labeled objects, with Bn := |Π(n)| denoting the nth Bell number. We note that exp(ez − 1) is the exponential generating function (GF) of the class of set partitions Πnk=1aXk k(λ), λ∈Π(n) and Xk(λ) is the number of blocks in λ of fixed size k, λ denoting a set partition. The series is computed in terms of the variable of maximum order, the coefficients of which are given in terms of the next-to-maximum order, etc This is more precise than mixing different terms. Our paper is organized as follows: in Section 2, we consider the central region, where we re-derive, with more precision, the asymptotic Gaussian property of Jn. In Section 3, we analyze the large deviation m = n/k − nα, with 1 > α > 1/2. We provide a brief justification of some integration procedures
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
