Explicit (Polynomial!) Expressions for the Expectation, Variance and Higher Moments of the Size of a (2n + 1, 2n + 3)-core partition with Distinct Parts

Abstract

Inspired by Armin Straub's conjecture (arXiv:1601.07161) about the number and maximal size of (2n+1, 2n+3)-core partitions with distinct parts, we develop relatively efficient, symbolic-computational algorithms, based on non-linear functional recurrences, to generate the generating functions, according to size, of the set of such partitions. By computing these polynomials for n=1,...21, we are able to rigorously derive explicit expressions for the expectation, variance, and third through seventh moments of the random variable "size of a (2n+1, 2n+3)-core partition with distinct parts." In particular, we find that this random variable is not asymptotically normal as n goes to infinity.

arXiv comment20 pages, 4 figures

References in corpus (4)

• Theorems, Problems and Conjectures
• Core partitions with distinct parts
• HISTABRUT: A Maple Package for Symbol-Crunching in Probability theory
• On $(2k+1,2k+3)$-core partitions with distinct parts

Cited by in corpus (4)

• Explicit expressions for the moments of the size of an (s,s+1)-core partition with distinct parts
• Core partitions with distinct parts
• A bijective proof of Amdeberhan's conjecture on the number of $(s,s+2)$-core partitions with distinct parts
• On the largest sizes of certain simultaneous core partitions with distinct parts