Fixed-perimeter tipping point at the critical modulus

Determine the limiting behavior of h(a)(n)-e(m1,m2)(n) for positive integers 0<m1<m2m, 0<ad, in the critical case m=2d+2 outside the cases resolved by Proposition 1.6.

Background

The paper studies fixed-perimeter analogues of the asymptotic comparison between d-distinct partitions and partitions whose parts lie in two residue classes. The functions h(a)(n) and e(m1,m2)(n) count fixed-perimeter partitions subject to d-distinctness and residue-class conditions, respectively.

The authors conjecture a limiting dichotomy when m is greater than or less than 2d+2, and Proposition 1.6 resolves several cases at the boundary m=2d+2. They explicitly leave the remaining boundary cases unresolved, stating that the limiting behavior is unclear based on their experiments.