Combinatorial Interpretation of an $A_{2n}^{(2)}$ Rogers–Ramanujan Identity

Definitions and mathematical statement

A partition is a finite weakly decreasing sequence of positive integers, including the empty partition. For a partition $\lambda$, write $|\lambda|=\sum_i\lambda_i$, set $2\lambda=(2\lambda_1,2\lambda_2,\ldots)$, and take $\lambda_1=0$ for the empty partition. Let $P_\mu(x_1,x_2,\ldots;t)$ denote the Hall--Littlewood symmetric function in the usual $P$-normalization. Thus the notation in the supplied statement uses an infinite principal specialization of a symmetric function.

Define \[ (a;q)_\infty=\prod_{r\geq0}(1-aq^r),\qquad \theta(a_1,\ldots,a_k;q)=\prod_{s=1}^k(a_s;q)_\infty(q/a_s;q)_\infty. \] For every pair of positive integers $m,n$, put $\kappa=2m+2n+1$. The requested problem is to find a combinatorial interpretation of the identity \[ \sum_{\substack{\lambda\text{ a partition}\\\lambda_1\leq m}} q^{|\lambda|}P_{2\lambda}(1,q,q^2,\ldots;q^{2n-1}) = \frac{(q^\kappa;q^\kappa)_\infty^n}{(q;q)_\infty^n} \prod_{i=1}^n\theta(q^{i+m};q^\kappa) \prod_{1\leq i<j\leq n}\theta(q^{j-i},q^{i+j-1};q^\kappa). \] Empty products are understood to equal $1$.

Unresolved qualifications. The supplied record does not specify the domain of $q$ or whether the identity is intended formally or analytically. Under a formal reading, the infinite specialization and products are interpreted coefficientwise in $q$; an analytic reading requires an explicitly stated domain and convergence conventions. The record also does not specify what form of combinatorial interpretation is required. Its broader original question asks for interpretations of the displayed $A_{2n}^{(2)}$ analogues of the Rogers--Ramanujan identities and other identities in the same family. Only the identity above is explicitly supplied, so the additional identities cannot be stated precisely from this record.

Short English statement

For every pair of positive integers $m,n$, find a combinatorial interpretation of the displayed Hall--Littlewood identity of type $A_{2n}^{(2)}$. The source also requests interpretations of other identities in the family, but does not supply their formulas here. The domain of $q$ and the precise intended form of interpretation remain unspecified. The supplied partially solved status is dated curation, not an independently verified conclusion.

Sources

• Supplied local problem registry record: permanent UnsolvedMath ID $30002546$, source problem code catalogue entry, entitled "Combinatorial Interpretation of an $A_{2n}^{(2)}$ Rogers--Ramanujan Identity." The record gives proposed year $2014$ and names no individual proposer.
• Enumerative Combinatorics (2014), Oberwolfach Reports, pp. 635--720, DOI 10.4171/owr/2014/12. Supplied report link: https://doi.org/10.4171/owr/2014/12. The record describes Oberwolfach Reports as documenting workshops and their open-problem discussions. It labels the statement corrected_verified and reports that it was checked against the official report, with hypotheses and notation restored while preserving the open questions. These are supplied verification claims; no independent source verification is asserted here.
• Supplied literature curation, checked 2026-08-21: status "partially solved." Its assessment states that combinatorial interpretations and proofs exist for classical and low-rank $A_2$ Rogers--Ramanujan families, while the full $A_{2n}^{(2)}$ family and the other identities in the extracted statement still lack a complete interpretation. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its status and cited claims require independent verification.
• Warnaar (2023), supplied literature reference: https://doi.org/10.1090/btran/147.
• "Cylindric A2 identities," supplied literature reference: https://arxiv.org/abs/2011.12828.
• Corteel (2017), supplied literature reference: https://doi.org/10.1090/proc/13373.