Theoretical Statistics and Mathematics Unit, ISI Delhi

Multiplicity of Summands in the Random Partitions of an Integer

by Ghurumuruhan Ganesan

In this paper, we prove a conjecture of Yakubovich regarding limit shapes of ``slices" of two-dimensional (2D) integer partitions and compositions of \(n\) when the number of summands \(m \sim An^{\alpha}\) for some \(A > 0\) and \(\alpha < \frac{1}{2}.\) We prove that the probability that there is a summand of multiplicity \(j\) in any randomly chosen partition or composition of an integer \(n\) goes to zero asymptotically with \(n\) provided \(j\) is larger than a critical value. As a corollary, we strengthen a result due to Erd$\ddot{o}$s and Lehner~\cite{erdos} that concerns the relation between the number of integer partitions and compositions when \(\alpha = \frac{1}{3}.\)

isid/ms/2012/12 [fulltext]

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