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Asymptotic analysis of expectations of plane partition statistics

  • Ljuben Mutafchiev
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Abstract

Assuming that a plane partition of the positive integer n is chosen uniformly at random from the set of all such partitions, we propose a general asymptotic scheme for the computation of expectations of various plane partition statistics as n becomes large. The generating functions that arise in this study are of the form Q(x)F(x), where \(Q(x)=\prod _{j=1}^\infty (1-x^j)^{-j}\) is the generating function for the number of plane partitions. We show how asymptotics of such expectations can be obtained directly from the asymptotic expansion of the function F(x) around \(x=1\). The representation of a plane partition as a solid diagram of volume n allows interpretations of these statistics in terms of its dimensions and shape. As an application of our main result, we obtain the asymptotic behavior of the expected values of the largest part, the number of columns, the number of rows (that is, the three dimensions of the solid diagram) and the trace (the number of cubes in the wall on the main diagonal of the solid diagram). Our results are similar to those of Grabner et al. (Comb Probab Comput 23:1057–1086, 2014) related to linear integer partition statistics. We base our study on the Hayman’s method for admissible power series.

Keywords

Plane partition statistic Asymptotic behavior Expectation 

Mathematics Subject Classification

05A17 05A16 11P82 

Notes

Acknowledgements

I am grateful the referee for the carefully reading the paper and for her/his helpful comments.

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Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.American University in BulgariaBlagoevgradBulgaria
  2. 2.Institute of Mathematics and Informatics of the Bulgarian Academy of SciencesSofiaBulgaria

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