MacMahon exponent-error factorization

Prove that for every m≥6 the exponent discrepancy \varepsilon_m^n between the actual and MacMahon-predicted product exponents factors as \varepsilon_m^n=\binom{n}{4}r_m(n) for an irreducible polynomial r_m of degree at most m−6.

Background

The paper defines the actual product exponents \omega_mn associated with the generating function of higher-dimensional partitions and compares them with MacMahon’s predicted exponents \overline{\omega}_mn. Their difference \varepsilon_mn is known to be a polynomial in n of degree at most m−1. The authors conjecture a stronger divisibility and degree statement, verified computationally through m=30.

References

For every $m\geqslant 6 $ there exists an irreducible polynomial $r_m(t)\in [t]$ of degree at most $m-6$ such that

\varepsilon_mn=\binom{n}{4}r_m(n)

for all $n\geqslant 1$. In particular, the error $\varepsilon_mn$ has degree at most $m-2$.

— Enumeration of partitions via socle reduction  (2501.10267 - Graffeo et al., 17 Jan 2025) in Conjecture 2, Section “Conjectures,” subsection “MacMahon’s discrepancy”