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

Yıl 2017, , 75 - 91, 11.01.2017
https://doi.org/10.13069/jacodesmath.37019

Öz

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.

Kaynakça

  • [1] B. Chern, P. Diaconis, D. M. Kane, R. C. Rhoades, Closed expressions for set partition statistics, Res. Math. Sci. 1(2) (2014) 1–32.
  • [2] B. Chern, P. Diaconis, D. M. Kane, R. C. Rhoades, Central limit theorems for some set partitions, Adv. Appl. Math. 70 (2015) 92–105.
  • [3] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, D. E. Knuth, On the LambertW function, Adv. Comput. Math. 5 (1996) 329–359.
  • [4] P. Flajolet, R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009.
  • [5] B. Fristedt, The structure of random partitions of large sets, Technical report, University of Minnesota, 1987.
  • [6] I. J. Good, Saddle–point methods for the multinomial distribution, Ann. Math. Statist. 28(4) (1957) 861–881.
  • [7] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Second Edition, Addison Wesley, 1994.
  • [8] H. K. Hwang, On convergence rates in the central limit theorems for combinatorial structures, European J. Combin. 19(3) (1998) 329–343.
  • [9] D. E. Knuth, The Art of Computer Programming, vol. 4a: Combinatorial Algorithms. Part I, Addison–Wesley, Upper Saddle River, New Jersey, 2011.
  • [10] G. Louchard, Asymptotics of the Stirling numbers of the first kind revisited: A saddle point approach, Discrete Math. Theor. Comput. Sci. 12(2) (2010) 167–184.
  • [11] G. Louchard, Asymptotics of the Stirling numbers of the second kind revisited: A saddle point approach, Appl. Anal. Discrete Math. 7(2) (2013) 193–210.
  • [12] G. Louchard, Asymptotics of the Eulerian numbers revisited: A large deviation analysis, Online J. Anal. Comb. 10 (2015) 1–11.
  • [13] T. Mansour, Combinatorics of Set Partitions, Discrete Mathematics and Its Applications Series, CRC Press, Boca Raton, FL, 2013.
  • [14] B. Salvy, J. Shackell, Symbolic asymptotics: Multiseries of inverse functions, J. Symbolic Comput. 20(6) (1999) 543–563.
  • [15] R. P. Stanley, Enumerative Combinatorics, Volume 1, 2nd edn, Cambridge Studies in Advanced Mathematics, Vol. 49. Cambridge University Press, Cambridge, 2012.
Yıl 2017, , 75 - 91, 11.01.2017
https://doi.org/10.13069/jacodesmath.37019

Öz

Kaynakça

  • [1] B. Chern, P. Diaconis, D. M. Kane, R. C. Rhoades, Closed expressions for set partition statistics, Res. Math. Sci. 1(2) (2014) 1–32.
  • [2] B. Chern, P. Diaconis, D. M. Kane, R. C. Rhoades, Central limit theorems for some set partitions, Adv. Appl. Math. 70 (2015) 92–105.
  • [3] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, D. E. Knuth, On the LambertW function, Adv. Comput. Math. 5 (1996) 329–359.
  • [4] P. Flajolet, R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009.
  • [5] B. Fristedt, The structure of random partitions of large sets, Technical report, University of Minnesota, 1987.
  • [6] I. J. Good, Saddle–point methods for the multinomial distribution, Ann. Math. Statist. 28(4) (1957) 861–881.
  • [7] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Second Edition, Addison Wesley, 1994.
  • [8] H. K. Hwang, On convergence rates in the central limit theorems for combinatorial structures, European J. Combin. 19(3) (1998) 329–343.
  • [9] D. E. Knuth, The Art of Computer Programming, vol. 4a: Combinatorial Algorithms. Part I, Addison–Wesley, Upper Saddle River, New Jersey, 2011.
  • [10] G. Louchard, Asymptotics of the Stirling numbers of the first kind revisited: A saddle point approach, Discrete Math. Theor. Comput. Sci. 12(2) (2010) 167–184.
  • [11] G. Louchard, Asymptotics of the Stirling numbers of the second kind revisited: A saddle point approach, Appl. Anal. Discrete Math. 7(2) (2013) 193–210.
  • [12] G. Louchard, Asymptotics of the Eulerian numbers revisited: A large deviation analysis, Online J. Anal. Comb. 10 (2015) 1–11.
  • [13] T. Mansour, Combinatorics of Set Partitions, Discrete Mathematics and Its Applications Series, CRC Press, Boca Raton, FL, 2013.
  • [14] B. Salvy, J. Shackell, Symbolic asymptotics: Multiseries of inverse functions, J. Symbolic Comput. 20(6) (1999) 543–563.
  • [15] R. P. Stanley, Enumerative Combinatorics, Volume 1, 2nd edn, Cambridge Studies in Advanced Mathematics, Vol. 49. Cambridge University Press, Cambridge, 2012.
Toplam 15 adet kaynakça vardır.

Ayrıntılar

Konular Mühendislik
Bölüm Makaleler
Yazarlar

Guy Louchard Bu kişi benim

Yayımlanma Tarihi 11 Ocak 2017
Yayımlandığı Sayı Yıl 2017

Kaynak Göster

APA Louchard, G. (2017). Multivariate asymptotic analysis of set partitions: Focus on blocks of fixed size. Journal of Algebra Combinatorics Discrete Structures and Applications, 4(1), 75-91. https://doi.org/10.13069/jacodesmath.37019
AMA Louchard G. Multivariate asymptotic analysis of set partitions: Focus on blocks of fixed size. Journal of Algebra Combinatorics Discrete Structures and Applications. Ocak 2017;4(1):75-91. doi:10.13069/jacodesmath.37019
Chicago Louchard, Guy. “Multivariate Asymptotic Analysis of Set Partitions: Focus on Blocks of Fixed Size”. Journal of Algebra Combinatorics Discrete Structures and Applications 4, sy. 1 (Ocak 2017): 75-91. https://doi.org/10.13069/jacodesmath.37019.
EndNote Louchard G (01 Ocak 2017) Multivariate asymptotic analysis of set partitions: Focus on blocks of fixed size. Journal of Algebra Combinatorics Discrete Structures and Applications 4 1 75–91.
IEEE G. Louchard, “Multivariate asymptotic analysis of set partitions: Focus on blocks of fixed size”, Journal of Algebra Combinatorics Discrete Structures and Applications, c. 4, sy. 1, ss. 75–91, 2017, doi: 10.13069/jacodesmath.37019.
ISNAD Louchard, Guy. “Multivariate Asymptotic Analysis of Set Partitions: Focus on Blocks of Fixed Size”. Journal of Algebra Combinatorics Discrete Structures and Applications 4/1 (Ocak 2017), 75-91. https://doi.org/10.13069/jacodesmath.37019.
JAMA Louchard G. Multivariate asymptotic analysis of set partitions: Focus on blocks of fixed size. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4:75–91.
MLA Louchard, Guy. “Multivariate Asymptotic Analysis of Set Partitions: Focus on Blocks of Fixed Size”. Journal of Algebra Combinatorics Discrete Structures and Applications, c. 4, sy. 1, 2017, ss. 75-91, doi:10.13069/jacodesmath.37019.
Vancouver Louchard G. Multivariate asymptotic analysis of set partitions: Focus on blocks of fixed size. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4(1):75-91.