Recursive socle reduction beyond degree three

Construct a recursive structure analogous to the recursion for M-partitions that reduces enumeration to partitions with socle degree at least 4.

Background

The paper’s socle-reduction strategy successfully reduces the enumeration of general higher-dimensional partitions to M-partitions, whose socles are constrained to lie in degrees at least 3. The authors explain that the same principle could be extended to impose an arbitrary lower bound on the socle degree, but they identify the next case—socle degree at least 4—as already lacking a comparable recursive mechanism.

References

However, already in the case of constraining the socle to be in degree at least 4, finding a recursive structure analogous to the one of \Cref{prop: intro alpha} is a highly nontrivial problem, which we leave to future exploration.

— Enumeration of partitions via socle reduction  (2501.10267 - Graffeo et al., 17 Jan 2025) in Section 1, Introduction, subsection “Refined partition counting: socle reduction technique”