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}.

翻译：我们提出了一种高效计算与Kostka系数相关的Gelfand--Tsetlin多面体Ehrhart多项式的方法。核心思想是利用Ehrhart–Macdonald互反性：在负整数点处评估Ehrhart多项式可转化为对严格Gelfand–Tsetlin模式的计数，而低膨胀系数下这类模式往往为零或数值极小。通过结合自适应策略，在每一步选择计算成本最低的评估点（正或负整数），该方法相较于通用多面体计算软件实现了显著的加速效果。我们以$\mathtt{OSCAR}$/$\mathtt{polymake}$为基准进行测试，并通过序多面体和置换偏序集案例展示了该方法的广泛适用性。相关实现已集成至Rust语言\texttt{kostka}包中，其优化方案也同步应用于\texttt{lrcalc-rs}（\texttt{lrcalc}的替代版本）。