Large parts of random plane partitions: a Poisson limit theorem. (English) Zbl 1494.05010
In this paper, a Poisson limit theorem for the number of large parts in a random plane partition is proven that parallels a similar result for ordinary integer partitions.
Consider a uniformly random plane partition of \(n\). It is shown that the number of parts that are greater than \(m\) asymptotically follows a Poisson distribution with mean \(\frac23 e^{-c}\) as \(n \to \infty\) if \[ m = \Big( \frac{n}{2\zeta(3)} \Big)^{1/3} \Big( \log \Big( \frac{n}{2\zeta(3)} \Big)^{2/3} + \log \log n + c \Big). \] The proof is based on generating functions and the saddle point method. The Poisson limit theorem is first proven for an auxiliary quantity whose difference from the number of parts greater than \(m\) is then shown to be negligible.
Consider a uniformly random plane partition of \(n\). It is shown that the number of parts that are greater than \(m\) asymptotically follows a Poisson distribution with mean \(\frac23 e^{-c}\) as \(n \to \infty\) if \[ m = \Big( \frac{n}{2\zeta(3)} \Big)^{1/3} \Big( \log \Big( \frac{n}{2\zeta(3)} \Big)^{2/3} + \log \log n + c \Big). \] The proof is based on generating functions and the saddle point method. The Poisson limit theorem is first proven for an auxiliary quantity whose difference from the number of parts greater than \(m\) is then shown to be negligible.
Reviewer: Stephan Wagner (Uppsala)
MSC:
| 05A17 | Combinatorial aspects of partitions of integers |
| 05A18 | Partitions of sets |
| 05A15 | Exact enumeration problems, generating functions |
| 62E15 | Exact distribution theory in statistics |
References:
| [1] | M. Abramovitz and I. A. Stegun,Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover Publ. Inc., New York, 1965. |
| [2] | G. E. Andrews,The Theory of Partitions, Encyclopedia Math. Appl.2, AddisonWesley, Reading, MA, 1976. · Zbl 0371.10001 |
| [3] | E. A. Bender and D. E. Knuth, Enumeration of plane partitions,J. Comb. Theory13(1972), 40-54. · Zbl 0246.05010 |
| [4] | P. Erd˝os and J. Lehner, The distribution of the number of summands in the partitions of a positive integer,Duke Math. J.8(1941), 335-345. · Zbl 0025.10703 |
| [5] | S. Finch,Mathematical Constants, Cambridge Univ. Press, Cambridge, 2003. · Zbl 1054.00001 |
| [6] | P. Flajolet and R. Sedgewick,Analytic Combinatorics, Cambridge Univ. Press, Cambridge, 2009. · Zbl 1165.05001 |
| [7] | B. Fristedt, The structure of random partitions of large integers,Trans. Amer. Math. Soc.337(1993), 703-735. · Zbl 0795.05009 |
| [8] | P. Grabner, A. Knopfmacher and S. Wagner, A general asymptotic scheme for the analysis of partition statistics,Combin. Probab. Comput.23(2014), 1057- 1086. · Zbl 1304.11123 |
| [9] | G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proc. London Math. Soc.17(2) (1918), 75-115. · JFM 46.0198.04 |
| [10] | W. K. Hayman, A generalization of Stirling’s formula,J. Reine Angew. Math. 196(1956), 67-95. · Zbl 0072.06901 |
| [11] | E. P. Kamenov and L. R. Mutafchiev, The limiting distribution of the trace of a random plane partition,Acta Math. Hungar.117(2007), 293-314. · Zbl 1250.05021 |
| [12] | D. E. Knuth, Permutations, matrices and generalized Young tableaux,Pacific J. Math.34(1970), 709-727. · Zbl 0199.31901 |
| [13] | P. A. MacMahon, Memoir on theory of partitions of Numbers VI: Partitions in two-dimensional space, to which is added an adumbration of the theory of partitions in three-dimensional space,Phil. Trans. Roy. Soc. London Ser. A211 (1912), 345-373. · JFM 42.0237.01 |
| [14] | P. A. MacMahon,Combinatory Analysis, Vol. 2, Cambridge Univ. Press, Cambridge (1916); reprinted by Chelsea, New York, 1960. · JFM 46.0118.07 |
| [15] | L. Mutafchiev, Asymptotic analysis of plane partition statistics,Abh. Math. Semin. Univ. Hamburg88(2018), 255-272. · Zbl 1395.05021 |
| [16] | L. Mutafchiev and E. Kamenov, Asymptotic formula for the number of plane partitions of positive integers,C. R. Acad. Bulgare Sci.59(2006), 361-366. · Zbl 1101.05009 |
| [17] | B. Pittel, On a likely shape of the random Ferrers diagram,Adv. Appl. Math. 18(1997), 432-488. · Zbl 0894.11039 |
| [18] | H. Rademacher, On the partition functionp(n),Proc. London Math. Soc.43 (1937), 241-254. · Zbl 0017.05503 |
| [19] | R. P. Stanley, Theory and applications of plane partitions I, II,Studies Appl. Math.50(1971), 156-188, 259-279. · Zbl 0225.05012 |
| [20] | R. P. Stanley, The conjugate trace and trace of a plane partition,J. Combin. Theory Ser. A14(1973), 53-65. · Zbl 0251.05006 |
| [21] | M. Szalay and P. Tur´an, On some problems of the statistical theory of partitions with applications to characters of the symmetric group, I,Acta Math. Acad. Sci. Hungar.29(1977), 361-379. · Zbl 0371.10033 |
| [22] | E. M. Wright, Asymptotic partition formulae, I: Plane partitions,Quart. J. Math. Oxford Ser. (2)2(1931), 177-189. · Zbl 0002.38202 |
| [23] | A. Young, On quantitative substitutional analysis,Proc. Lond. Math, Soc.33 (1901), 97-146 · JFM 32.0157.02 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
