An Arithmetic Property of Moments of the $\beta$-Hermite Ensemble and Certain Map Enumerators

Moments of the $\beta$-Hermite ensemble are known to be related to the enumerative theory of topological maps. When $\beta \in \{ 1, 2 \}$, asymptotic information about these moments has been used to deduce asymptotics on the number of maps of given genus, and arithmetic information about these moments can sometimes be explained by underlying group actions on the set of maps. In this paper we establish a new arithmetic property about the $2q$-th moment of the $\beta$-Hermite ensemble, for any prime $q \ge 3$ and real number $\beta > 0$, that has a combinatorial interpretation in terms of maps but no known combinatorial explanation. In the process, we derive several additional results that might be of independent interest, including a general integrality statement and an efficient algorithm for evaluating expectations of multi-part elementary symmetric polynomials of bounded length.