Moments about the mean of the size of a self-conjugate (s,t)-core partition

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.