Optimization over Nonzero Littlewood–Richardson Coefficients

Definitions and mathematical statement

A partition $\alpha$ is a finite weakly decreasing sequence of positive integers. Write $|\alpha|=\sum_i\alpha_i$ for its size and $\ell(\alpha)$ for its length. For a nonempty partition $\alpha$, put $m=|\alpha|$, $r=\ell(\alpha)$, and define \[ \eta(\alpha)=m^2+\frac{r(r-1)(2r-1)}{6}-\sum_{i=1}^{r}\left(\alpha_i-(i-1)\right)^2. \] Thus, in evaluating $\eta(\mu)$ and $\eta(\nu)$, the size and length are those of the respective partition. The usual empty-partition and empty-sum conventions would extend this definition by $\eta(\varnothing)=0$; the supplied record does not explicitly specify whether empty partitions are admitted.

For partitions $\lambda,\mu,\nu$, let $c^{\lambda}_{\mu,\nu}$ denote the Littlewood--Richardson coefficient, defined by the Schur-function expansion \[ s_\mu s_\nu=\sum_\lambda c^{\lambda}_{\mu,\nu}s_\lambda. \] Define \[ \Delta(\lambda,\mu,\nu)=\eta(\lambda)-\eta(\mu)-\eta(\nu). \] The supplied problem asks for the exact maximum of $\Delta$, given $n,k,d$ with $k\le n/2$, subject to \[ |\lambda|=n,\qquad |\mu|=k,\qquad |\nu|=n-k,\qquad c^{\lambda}_{\mu,\nu}\ne0. \] The size conditions require nonnegative integer sizes, and $d$, as a partition length, must be an integer. However, the source does not explicitly give the parameter ranges or boundary conventions.

Unresolved length qualification. The introductory definition describes $\lambda$ as having length $d$, but the subsequent list of optimization constraints does not explicitly repeat a length condition. Under the reading that this introductory condition remains in force, the requested quantity is \[ M(n,k,d)=\max\left\{\Delta(\lambda,\mu,\nu): \begin{array}{l} \lambda,\mu,\nu\text{ are partitions},\quad \ell(\lambda)=d,\\ |\lambda|=n,\quad |\mu|=k,\quad |\nu|=n-k,\\ c^{\lambda}_{\mu,\nu}\ne0 \end{array}\right\}, \] for parameters with a nonempty feasible set. This is a conditional formulation requiring review: the supplied record does not explicitly resolve whether the intended restriction is exactly $\ell(\lambda)=d$, a bound on lengths, or only the displayed size and nonvanishing constraints. With only the latter constraints, the maximum has no dependence on $d$. No value is specified for an empty feasible set.

Short English statement

Determine the largest value of $\eta(\lambda)-\eta(\mu)-\eta(\nu)$ among partitions of the prescribed sizes whose Littlewood--Richardson coefficient is nonzero. The role of the given length parameter $d$, the permitted boundary cases, and empty feasible sets require clarification. The supplied record labels the problem open; this draft does not independently verify that status.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30006533$, source problem code catalogue entry, title "Optimization over Nonzero Littlewood--Richardson Coefficients." The supplied original excerpt labels this as problem 10.
• Supplied primary citation: Enumerative Combinatorics (2026), Oberwolfach Reports, report 2026/2, pp. 81--156, DOI 10.4171/owr/2026/2. https://doi.org/10.4171/owr/2026/2. The excerpt attributes the problem to Matjaž Konvalinka and says it was communicated by Igor Klep at the University of Ljubljana.
• Supplied literature source: Taskin--Gumus--Isik--Ulvi, Lower and Upper Bounds for Nonzero Littlewood--Richardson Coefficients. https://arxiv.org/abs/2208.06541.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that existing work gives criteria and bounds for nonvanishing but no exact solution to this optimization. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry supplies the statement status corrected_verified, explaining that only formula delimiters, notation, grammar, and typography were repaired while preserving mathematical content and scope. This supplied editorial label does not constitute mathematical verification of this draft.