For every pair of coprime integers 1<a<b, the matrix-counting q-series Z_a,b(q) equals the explicit infinite product P_a,b(q), namely Z_a,b(q)=P_a,b(q). Equivalently, for every prime power q, with NC_n^a,b( F_q) the set of pairs of commuting nilpotent matrices (A,B)∈ M_n( F_q)^2 satisfying A^a=B^b, one has displaystyle ∏_m≥ 1(1-q^-m)∑_n=0^∞frac|NC_n^a,b( F_q)||operatornameGL_n( F_q)|=Z_a,b(q^-1)=P_a,b(q^-1).

Huang–Jiang–Oblomkov conjecture

For every pair of coprime integers $1<a<b$ , the matrix-counting $q$ -series $Z_{a,b}(q)$ equals the explicit infinite product $P_{a,b}(q)$ , namely $Z_{a,b}(q)=P_{a,b}(q)$ . Equivalently, for every prime power $q$ , with $\mathcal{NC}_n^{a,b}(\mathbb F_q)$ the set of pairs of commuting nilpotent matrices $(A,B)\in M_n(\mathbb F_q)^2$ satisfying $A^a=B^b$ , one has $\displaystyle \prod_{m\geq 1}(1-q^{-m})\sum_{n=0}^{\infty}\frac{|\mathcal{NC}_n^{a,b}(\mathbb F_q)|}{|\operatorname{GL}_n(\mathbb F_q)|}=Z_{a,b}(q^{-1})=P_{a,b}(q^{-1})$ .

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Geometric point-count formulation
For every pair of coprime integers $1<a<b$ and every prime power $q$ , the normalized generating series of commuting nilpotent matrix pairs satisfying $A^a=B^b$ equals the HJO $q$ -series and the explicit product: $\displaystyle \prod_{m\geq 1}(1-q^{-m})\sum_{n=0}^{\infty}\frac{|\mathcal{NC}_n^{a,b}(\mathbb F_q)|}{|\operatorname{GL}_n(\mathbb F_q)|}=Z_{a,b}(q^{-1})=P_{a,b}(q^{-1})$ .
source: Rogers--Ramanujan identities from the geometry of $X^a=Y^b$

References

Primary source

arXiv

Additional references

Rogers--Ramanujan identities from the geometry of X^a=Y^b — arXiv — Yifeng Huang, Kenny Lau, Ken Ono

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed progress

A new result proves all cases with first parameter three, while the general conjecture remains open.

The conjecture asserts an equality between a matrix-counting $q$ -series $Z_{a,b}(q)$ and an explicit product $P_{a,b}(q)$ for coprime $a,b>1$ . The general statement is not yet proved.

Known results

The $a=2$ layer is classical, following from the Andrews–Gordon identities.
Huang, Jiang, and Oblomkov proved $(a,b)=(3,4),(3,5),(3,7),(3,8)$ .
Lau and Ono subsequently proved the stronger identity for every $b>3$ coprime to $3$ , establishing the full $a=3$ layer.

2026 finite-identity advance

A stronger finite identity links commuting nilpotent matrix-pair counts, $Z_{a,b}(q)$ , and cylindric partitions; its limit yields the proved $a=3$ cases and a new infinite family of Rogers–Ramanujan-type identities. AxiomProver generated a Lean certificate conditional on cited literature, so this is not independent verification of the mathematics.

Current status (as of September 2026): The classical $a=2$ layer and the entire $a=3$ layer are claimed proved, while the conjecture for general $a>3$ remains open.

AxiomProverAxiomProversolved2026-09-17evidence
Huang–Jiang–Oblomkov Rogers–Ramanujan conjecture proved

Sources

No solutions have been posted yet.

Geometric point-count formulation

Primary source

Additional references

Progress summary