Let C_a,b,m(q) denote the infinite copartition-product generating function of Burson and Eichhorn, and let C_a,b,m;N,M(q) denote its finite version. The infinite conjecture asserts that, whenever b| a, every coefficient is non-negative: [q^n]C_a,b,m(q)≥ 0 for all n≥ 0. The finite conjecture asserts that, whenever a+b=m and N≤ M, every coefficient is non-negative: [q^n]C_a,b,m;N,M(q)≥ 0 for all n≥ 0. The supplied sources identify these copartition products and parameter conditions but do not give their explicit product formulas.

Burson–Eichhorn copartition-product non-negativity conjectures

Let $\mathcal{C}_{a,b,m}(q)$ denote the infinite copartition-product generating function of Burson and Eichhorn, and let $\mathcal{C}_{a,b,m;N,M}(q)$ denote its finite version. The infinite conjecture asserts that, whenever $b\mid a$ , every coefficient is non-negative: $[q^n]\mathcal{C}_{a,b,m}(q)\ge 0$ for all $n\ge 0$ . The finite conjecture asserts that, whenever $a+b=m$ and $N\le M$ , every coefficient is non-negative: $[q^n]\mathcal{C}_{a,b,m;N,M}(q)\ge 0$ for all $n\ge 0$ . The supplied sources identify these copartition products and parameter conditions but do not give their explicit product formulas.

References

Primary source

arXiv

Additional references

Further results on non-negativity conjectures for copartition products — arXiv — Haijun Li

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed progress

The conjectures remain open, but a September 2026 preprint claims several new parameter cases and a stronger initial positivity range.

The Burson–Eichhorn conjectures assert coefficientwise non-negativity for finite and infinite copartition-product generating functions under specified divisibility and truncation conditions. The infinite conjecture concerns parameters satisfying $b\mid a$ ; the finite conjecture additionally assumes $a+b=m$ and $N\leq M$ .

Known results

Burson–Eichhorn et al. (2022) proved the infinite case $a=b$ .
The same paper recorded Craig’s earlier special case $a=3$ , $b=1$ , $m=4$ .
The finite conjecture was reduced to a recursion and was shown to imply the infinite conjecture, but not proved in general.

September 2026 claimed extensions

Haijun Li’s preprint Further results on non-negativity conjectures for copartition products claims the finite conjecture when a truncation parameter is $1$ or residue parameters coincide, the infinite conjecture when $a\equiv b\pmod m$ , and uniform non-negativity through degrees below $a+2m$ . It also claims structural decompositions and eventual quasipolynomial behavior; these advances do not settle either conjecture in full.

Current status (as of September 2026): Several additional cases are claimed, but the finite and infinite conjectures remain open in general and the latest claims are unverified.

Sources

No solutions have been posted yet.