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.

Attempts by AI

AI prompts

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 ≤ 7 and sum of parts ≤ 30.

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

Mathematician survey

The author assessed the problem as follows.