Output-sensitive computation of skew Kostka numbers

Construct an algorithm that computes the skew Kostka number K_{λ/μ,ν} in time O(K_{λ/μ,ν} poly(n)) for all partitions λ, μ, and ν of compatible sizes.

Background

The paper interprets skew Kostka numbers as counts of integer points in Gelfand–Tsetlin polytopes and obtains polynomial-time algorithms when relevant partition lengths are fixed. It then seeks an output-sensitive algorithm whose running time is essentially linear in the value of the coefficient, which would be especially meaningful when the coefficient is polynomially large but the partition lengths are not constant. The proposed conjecture is noted to be vacuously true in many cases where the coefficient itself is exponentially large.

References

We pose the following conjecture which generalizes the result in [LH24]. Conjecture 4.1. Let λ, μ, ν ⊢ n. Then there exists an algorithm computing Kλ/μ,ν in timeO(Kλ/μ,ν poly(n)).

— Polynomial time classical versus quantum algorithms for representation theoretic multiplicities  (2502.20253 - Panova, 27 Feb 2025) in Conjecture 4.1, Section 4, page 10

This efficient algorithm does not apply when the lengths are not constant. We pose the following conjecture which generalizes the result in [LH24].Conjecture 4.1. Let λ, μ, ν ⊢ n. Then there exists an algorithm computing Kλ/μ,ν in timeO(Kλ/μ,ν poly(n)).

— Polynomial time classical versus quantum algorithms for representation theoretic multiplicities  (2502.20253 - Panova, 27 Feb 2025) in Conjecture 4.1, Section 4, page 10