Explicit expressions for the expectation, variance and higher moments of the size of a (2<i>n</i> + 1, 2<i>n</i> + 3)-core partition with distinct parts

Abstract

Armin Straub’s beautiful article (https://arxiv.org/abs/1601.07161) concludes with two intriguing conjectures about the number, and maximal size, of (2n+1,2n+3)-core partitions with distinct parts. These were proved by ingenious, but complicated, argumentsby Sherry H.F. Yan, Guizhi Qin, Zemin Jin, and Robin D.P. Zhou (https://arxiv.org/abs/1604.03729). In the present article, we first comment that these results can be proved faster by ‘experimental mathematics’ methods, that are easily rigorizable.We then develop relatively efficient, symbolic-computational, algorithms, based on non-linear functional recurrences,to generate what we call the Straub polynomials, where Sn(q) is the generating function, according to size,of the set of (2n+1,2n+3)-core partitions with distinct parts, and compute the first 21 of them.These are used to deduce explicit expressions, as polynomials in n, for the mean, variance, and the third through the seventh moments (about the mean) of the random variable ‘size’ defined on (2n+1,2n+3)-core partitions with distinct parts. In particular we show that this random variable is not asymptotically normal, and the limit of the coefficient of variation is 14010/150=0.789092305…,the scaled-limit of the third moment (skewness) is (396793/390815488)·467·7680=1.92278748…, and that the scaled-limit of the 4th-moment (kurtosis) is 145309380/16792853=8.6530490…. We are offering to donate one hundred dollars to the OEIS foundation in honor of the first to identify the limiting distribution.