Stretched Littlewood-Richardson Coefficients

Find partitions whose stretched LR-coefficients, when expressed as a polynomial, have a negative coefficient.

Notability: Major advance

Solved: No

Field: Combinatorics

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About the problem Attempts by AI AI prompts Mathematician survey

About the problem

The Littlewood-Richardson (LR) coefficients are central quantities in algebraic combinatorics, appearing in several interrelated contexts. They are indexed by partitions $\lambda, \mu, \nu$ and written $c^{\nu}_{\lambda \mu}$. The stretched LR-coefficients are the LR-coefficients of integer scalings of the underlying partitions, written $c^{t \nu}_{t \lambda, t \mu}$.

The stretched LR-coefficients are known to be polynomial in $t$. It is conjectured that the coefficients of this polynomial are positive, but the problem author expects this to be false. The aim of this problem is to find a counterexample.

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Attempts by AI

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GPT-5.2 Pro

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Gemini 3 Deep Think

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Full problem

Let c^{tλ}_{tμ,tν} denote the stretched Littlewood-Richardson coefficients associated with integer partitions λ, μ, ν, where |λ| = |μ| + |ν|. Find integer partitions λ, μ, ν, such that when c^{tλ}_{tμ,tν} is expressed as a polynomial in t, at least one of the coefficients of this polynomial is negative. Each of the partitions must have length ≤ 20 and largest part ≤ 50.

Produce your partitions as a list of Python lists of ints.

Mathematician survey

The author assessed the problem as follows.

# of mathematicians highly familiar with the problem: a majority of those working in a subfield (≈100)

# of mathematicians who have made a serious attempt to solve the problem: 5–10

Rough guess of how long it would take an expert human to solve the problem: 1–3 years

Notability of a solution: a major advance in a subfield

A solution would be published: in a leading general journal

Likelihood of a solution generating more interesting math: somewhat likely: an especially innovative solution could, but is hardly guaranteed

Probability that the problem is solvable as stated: 60-80%