Andrews conjecture on refined MacMahon coefficients

Prove that, for every positive integer k, the generating function of the refined coefficients \overline{y}_{i+k+2}^{i+1} is a rational function with numerator degree at most \binom{k+4}{2}−7−k and denominator \prod_{i=1}^k\prod_{j=1}^i(1−jt), and establish the nonnegativity \overline{y}_d^k≥0 for all 0≤k≤d−1.

Background

The authors compare the actual higher-dimensional partition numbers with the coefficients predicted by MacMahon’s product formula. They introduce refined coefficients \overline{y}_dk for the product-formula numbers \pi_dn and formulate a stronger conjecture asserting both rationality of a family of generating functions and nonnegativity of all refined coefficients. Computational checks are reported for substantial ranges, but no proof is given.

References

For every $k \in {> 0}$ there exists a polynomial $\overline{\mathsf y}{k}(t)\in[t]$ of degree at most $\binom{k+4}{2}- 7-k$, such that

\sum_{i\geqslant 0} \overline{y}{i+k+2}{i+1} ti= \frac{ \overline{\mathsf y}{k}(t)}{ \prod_{i=1}k\prod_{j=1}i (1-jt)}.

Moreover, for every $0\leqslant k\leqslant d-1$ we have $\overline{y}_{d}k\geqslant 0$.

— Enumeration of partitions via socle reduction  (2501.10267 - Graffeo et al., 17 Jan 2025) in Conjecture 1, Section “Conjectures,” subsection “MacMahon’s discrepancy”