Fast computation of Ehrhart polynomials of Gelfand--Tsetlin polytopes via Macdonald reciprocity

Abstract

We describe an efficient method for computing the Ehrhart polynomial of Gelfand--Tsetlin polytopes arising from Kostka coefficients. The key idea is to exploit Ehrhart--Macdonald reciprocity: evaluating the Ehrhart polynomial at negative integers reduces to counting \emph{strict} Gelfand--Tsetlin patterns, which are often zero or very small for low dilations. Combined with an adaptive strategy that chooses the cheapest evaluation point (positive or negative) at each step, this yields substantial practical speedups compared to general-purpose polytope software. We benchmark against $\mathtt{OSCAR}$/$\mathtt{polymake}$, and illustrate the broader applicability of the method through order polytopes and permutation posets. The implementation is available in the Rust \texttt{kostka} package, with related optimizations also incorporated in the new \texttt{lrcalc-rs} replacement for \texttt{lrcalc}.