×

Phase transitions from \(\exp( n^{1/2})\) to \(\exp(n^{2/3})\) in the asymptotics of banded plane partitions. (English) Zbl 1458.05023

G. H. Hardy and S. Ramanujan [Proc. Lond. Math. Soc. (2) 17, 75–115 (1918; JFM 46.0198.04)] established the following well-known asymptotic formula for the number of partitions of \(n\): \[ p(n)=[q^n]\prod_{k=1}^\infty\dfrac{1}{1-q^k}\sim\dfrac{1}{4\sqrt{3}n}\exp{\left(\sqrt{\dfrac{2n}{3}}\pi\right)}. \]
E. M. Wright [Q. J. Math., Oxf. Ser. 2, 177–189 (1931; JFM 59.0201.01)] found the following asymptotic formula for the number of plane partitions of \(n\): \[ pp(n)=[q^n]\prod_{k=1}^\infty\dfrac{1}{(1-q^k)^k}\sim\dfrac{\zeta(3)^{7/36}e^{-\zeta'(-1)}}{2^{11/36}\sqrt{3\pi}n^{25/36}}\exp {\left(3\sqrt[3]{\dfrac{\zeta(3)n^2}{4}}\right)}. \] Here, the symbol \([q^n]f(q)\) denotes the coefficient of \(q^n\) in the Taylor expansion of \(f\) and \(\zeta(s)\) the Riemann zeta function.
The authors investigate the asymptotic behaviour of a class of banded plane partitions under a varying bandwidth parameter \(m\), and clarify the transitional behavior for large size \(n\) and increasing \(m=m(n)\) to be from \(c_1n^{-1}\exp{\left(c_2n^{1/2}\right)}\) to \(c_3n^{-49/72} \exp{\left(c_4n^{2/3}+c_5n^{1/3}\right)}\) for some explicit coefficients \(c_1,\ldots,c_5\). The main ingredient of proof is a unified saddle-point analysis for all phases. This method is general and can be extended to other classes of plane partitions.

MSC:

05A17 Combinatorial aspects of partitions of integers
11P82 Analytic theory of partitions
05A16 Asymptotic enumeration

Software:

OEIS
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Andrews, G. E., The Theory of Partitions (1976), Addison-Wesley Publishing Co.: Addison-Wesley Publishing Co. Reading, Mass.-London-Amsterdam · Zbl 0371.10001
[2] Apostol, T. M., Introduction to Analytic Number Theory (1976), Springer-Verlag: Springer-Verlag New York-Heidelberg · Zbl 0335.10001
[3] Ayoub, R., An Introduction to the Analytic Theory of Numbers (1963), American Mathematical Society: American Mathematical Society Providence, R.I. · Zbl 0128.04303
[4] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Higher Transcendental Functions. Volume I (1953), McGraw-Hill Book Company, Inc.: McGraw-Hill Book Company, Inc. New York-Toronto-London · Zbl 0051.30303
[5] Erdös, P.; Lehner, J., The distribution of the number of summands in the partitions of a positive integer, Duke Math. J., 8, 335-345 (1941) · JFM 67.0126.02
[6] Flajolet, P.; Gourdon, X.; Dumas, P., Mellin transforms and asymptotics: harmonic sums, Theor. Comput. Sci., 144, 1-2, 3-58 (1995) · Zbl 0869.68057
[7] Flajolet, P.; Sedgewick, R., Analytic Combinatorics (2009), Cambridge University Press · Zbl 1165.05001
[8] Gordon, B.; Houten, L., Notes on plane partitions. I, J. Comb. Theory, 4, 72-80 (1968); Gordon, B.; Houten, L., Notes on plane partitions. II, J. Comb. Theory, 4, 81-99 (1968) · Zbl 0153.32803
[9] Gordon, B.; Houten, L., Notes on plane partitions. III, Duke Math. J., 36, 801-824 (1969) · Zbl 0194.32302
[10] Han, G.-N.; Xiong, H., Some useful theorems for asymptotic formulas and their applications to skew plane partitions and cylindric partitions, Adv. Appl. Math., 96, 18-38 (2018) · Zbl 1383.05018
[11] Han, G.-N.; Xiong, H., Skew doubled shifted plane partitions: calculus and asymptotics (2019)
[12] Hardy, G. H., Divergent Series (1992), Clarendon Press: Clarendon Press Oxford · Zbl 0032.05801
[13] Hardy, G. H.; Ramanujan, S., Asymptotic formulæ in combinatory analysis, Proc. Lond. Math. Soc. (2), 2, 1, 75-115 (1918) · JFM 46.0198.04
[14] Kamenov, E.; Mutafchiev, L., The limiting distribution of the trace of a random plane partition, Acta Math. Hung., 117, 4, 293-314 (2007) · Zbl 1250.05021
[15] Knuth, D. E., Big omicron and big omega and big theta, ACM SIGACT News, 8, 2, 18-24 (1976)
[16] Meinardus, G., Asymptotische Aussagen über Partitionen, Math. Z., 59, 388-398 (1954) · Zbl 0055.03806
[17] Mutafchiev, L., The size of the largest part of random plane partitions of large integers, Integers, 6, A13 (2006) · Zbl 1091.05006
[18] OEIS Foundation Inc, The on-line encyclopedia of integer sequences. · Zbl 1439.11001
[19] Olver, F. W.J., Asymptotics and Special Functions (1974), Academic Press: Academic Press New York-London · Zbl 0303.41035
[20] Pittel, B., On dimensions of a random solid diagram, Comb. Probab. Comput., 14, 873-895 (2005) · Zbl 1080.05007
[21] Stanley, R. P., Theory and application of plane partitions. II, Stud. Appl. Math., 50, 259-279 (1971) · Zbl 0225.05012
[22] Whittaker, E. T.; Watson, G. N., A Course of Modern Analysis (1996), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0951.30002
[23] Wright, E. M., Asymptotic partition formulae: I. Plane partitions, Q. J. Math., 2, 1, 177-189 (1931) · Zbl 0002.38202
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.