Moments about the mean of the size of a self-conjugate [formula omitted]-core partition

Abstract

Johnson proved that if s,t are coprime integers, then the rth moment of the size of an (s,t)-core is a polynomial of degree 2r in t for fixed s. After that, by defining a statistic size on elements of affine Weyl group, which is preserved under the bijection between minimal coset representatives of S˜t∕St and t-cores, Thiel and Williams obtained the variance and the third moment about the mean of the size of an (s,t)-core. Later, Ekhad and Zeilberger stated the first six moments about the mean of the size of an (s,t)-core and the first nine moments about the mean of the size of an (s,s+1)-core using Maple. To get the moments about the mean of the size of a self-conjugate (s,t)-core, we proceed to follow the approach of Thiel and Williams, however, their approach does not seem to directly apply to the self-conjugate case. In this paper, following Johnson’s approach, by Ehrhart theory and Euler–Maclaurin theory, we prove that if s,t are coprime integers, then the rth moment about the mean of the size of a self-conjugate (s,t)-core is a quasipolynomial of period 2 and degree 2r in t for fixed odd s. Then, based on a bijection of Ford, Mai and Sze between self-conjugate (s,t)-cores and lattice paths in s2×t2 rectangle and a formula of Chen, Huang and Wang on the size of self-conjugate (s,t)-cores, we obtain the variance, the third moment and the fourth moment about the mean of the size of a self-conjugate (s,t)-core.