Seaweed Algebras and the Index Statistic for Partitions

Abstract

In 2018 Coll, Mayers, and Mayers conjectured that the $q$-series $(q,-q^3;q^4{)}_{\infty }^{-1}$ is the generating function for a certain parity statistic related to the index of seaweed algebras. We prove this conjecture. Thanks to earlier work by Seo and Yee, the conjecture would follow from the non-negativity of the coefficients of this infinite product. Using a variant of the circle method along with Euler-Maclaurin summation, we establish this non-negativity, thereby confirming the Coll-Mayers-Mayers Conjecture.

arXiv commentUpdated to reflect changes in certain numerical constants in results cited from another paper