For a permutation $u\in S_n$, let $N\ast u\in S_{Nn}$ be the permutation with scaled Lehmer code. For given $u,v,w\in S_n$ and integer $N$, the stretched Schubert coefficients are defined as $f_{u,v,w}(N):=c_{N*u,N*v}^{N*w}$. Our main result is that the function $f_{u,v,w}(N)$ is eventually quasi-polynomial. This proves Kirillov's conjecture (2004), that the generating function for the sequence $\{f_{u,v,w}(N)\}$ is rational. For the proof, we use combinatorics of pipe dreams to show that Schubert coefficients are given as an alternating sum of the numbers of integer points in certain polytopes. These polytopes behave nicely under stretching, and we use Ehrhart theory to obtain the result. As a consequence of the proof, we also present new counterexamples to the saturation conjecture for Schubert coefficients, and give computational applications.

翻译：对于置换$u\in S_n$，令$N\ast u\in S_{Nn}$为具有缩放Lehmer码的置换。给定$u,v,w\in S_n$及整数$N$，拉伸舒伯特系数定义为$f_{u,v,w}(N):=c_{N*u,N*v}^{N*w}$。我们的主要结果是函数$f_{u,v,w}(N)$最终为准多项式，从而证明了Kirillov（2004年）关于序列$\{f_{u,v,w}(N)\}$的生成函数为有理函数的猜想。在证明中，我们利用管道梦的组合学方法，将舒伯特系数表示为某些多胞体中整点数的交错和。这些多胞体在拉伸下具有良好的性质，进而借助Ehrhart理论得到结果。作为证明的推论，我们还给出了舒伯特系数饱和猜想的新反例，并提供了计算应用。