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

作者：Aggarwal Amol^*

来源：Electronic Journal of Combinatorics, 2018, 25(1): P1.29.

DOI：10.37236/6661

• Topological maps
• beta-Hermite ensemble
• elementary symmetric functions

Moments of the beta-Hermite ensemble are known to be related to the enumerative theory of topological maps. When beta is an element of {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 >= 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 multipart elementary symmetric polynomials of bounded length.

• 出版日期2018-2-16