Eventual quasi-polynomiality of simple and unrestricted column-number functions

Determine whether the counting functions \(\s(\Delta,r)\) and \(\h(\Delta,r)\) are eventually polynomial or quasi-polynomial when either the number of rows \(r\geq3\) is fixed or the modularity parameter \(\Delta\) is fixed.

Background

For two-row matrices, the paper establishes eventual quasi-linearity of $\g(\Delta,2)$, while related exact results imply polynomial behavior in several special cases. The authors seek to determine whether analogous eventual polynomial or quasi-polynomial behavior occurs for the simple and unrestricted column-number functions $\s$ and $\h$, particularly in settings with at least three rows.