Multivariate asymptotic analysis of set partitions: Focus on blocks of fixed size
Keywords:
Set partitions, Bell numbers, Asymptotics, Saddle point method, Multiseries expansions, Analytic combinatoricsAbstract
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.
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B. Chern, P. Diaconis, D. M. Kane, and R. C. Rhoades, Closed expressions for averages of set partition statistics, Res. Math. Sci. 1(1) (2014) Article no. 2.
B. Chern, P. Diaconis, D. M. Kane, and R. C. Rhoades, Central limit theorems for some set partition statistics, Adv. in Appl. Math. 70 (2015) 92–101.
R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, On the Lambert $W$ function, Adv. Comput. Math. 5(1) (1996) 329–359.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, Cambridge, 2009.
B. Fristedt, The structure of random partitions of large integers, Trans. Amer. Math. Soc. 337(2) (1993) 703–735.
I. J. Good, Saddle–point methods for the multinomial distribution, Ann. Math. Statist. 28(4) (1957) 861–881.
R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed., Addison-Wesley, Reading, MA, 1994.
H.-K. Hwang, On convergence rates in the central limit theorems for combinatorial structures, European J. Combin. 19(3) (1998) 329–343.
D. E. Knuth, The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Part 1, Addison-Wesley, Upper Saddle River, NJ, 2011.
G. Louchard, Asymptotics of the Stirling numbers of the first kind revisited: A large deviations approach, Discrete Math. Theor. Comput. Sci. 12(2) (2010) 93–112.
G. Louchard, Asymptotics of the Stirling numbers of the second kind revisited: A large deviations approach, Appl. Anal. Discrete Math. 7(2) (2013) 293–310.
G. Louchard, Asymptotics of the Eulerian numbers revisited: A large deviations approach, Open J. Discrete Appl. Math. 1(1) (2018) 1–17.
T. Mansour, Combinatorics of Set Partitions, Discrete Mathematics and Its Applications, CRC Press, Boca Raton, FL, 2012.
B. Salvy and J. Shackell, Symbolic asymptotics: Multiseries of inverse functions, J. Symbolic Comput. 27(6) (1999) 543–563.
