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由Kostka系数诱导的Gelfand--Tsetlin多面体Ehrhart多项式的方法。核心思路在于利用Ehrhart--Macdonald互反性：将Ehrhart多项式在负整数点处的求值转化为对\emph{严格}Gelfand--Tsetlin模式的计数，而低膨胀系数下此类模式通常为零或极小。结合自适应策略——每一步选取最优（正或负）求值点——该方法相较于通用多面体计算软件实现了显著的实践加速。我们以$\mathtt{OSCAR}$/$\mathtt{polymake}$为基准进行性能测试，并通过序多面体与排列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，其中关联优化亦被纳入新版\texttt{lrcalc-rs}（用以替代\texttt{lrcalc}）。