Seminar: The Asymptotic Analysis of Norms in Integer Partitions
Speaker: Deniis Kinoti Gikunda, Stellenboch University, South Africa
Abstract: A partition of a positive integer n is a way of expressing n as a sum of positive integers. In this talk, we focus on a specific parameter of partitions known as the norm, defined as the product of the parts in a partition. We show that the logarithm of the norm of a random partition of n converges to a continuous limiting distribution as n → ∞. We generalize this result to the case of Λ-partitions, where Λ is a sequence of positive integers satisfying certain analytic conditions (referred to as the Meinardus scheme). Important examples include ordinary partitions, square partitions, prime partitions, Mahler partitions, and plane partitions. Our methods rely on classical tools from analytic number theory and combinatorics, including the saddle-point method and Mellin transform techniques. We conclude with a discussion of potential directions for future research.
Keywords: Partitions, Mellin transform, limiting distribution, analytic number theory