We study exact uniform sampling of permutations of length $n$ whose longest increasing subsequence (LIS) has prescribed length $k$. For $k \in Θ(n)$, we give a direct rejection sampler whose expected running time is $O(n\log\log n)$ in the word-RAM model. The sampler uses an expanded proposal space consisting of permutations together with a specified increasing subsequence, and accepts exactly those proposals whose specified subsequence is the leftmost LIS. For arbitrary $1\le k\le n$, we give an exact sampler based on the Robinson--Schensted correspondence. The algorithm samples the corresponding Plancherel-conditioned shape by computing exact completion counts via determinant identities, and then samples two uniform tableaux of that shape. The direct implementation runs in $\tilde O(n^4k^5)$ expected time. We then show that the same sampler can be implemented in expected $\tilde O(n^3k^4)$ time by evaluating a determinant oracle through Hankel moment matrices.

翻译：我们研究了长度为 $n$ 且最长递增子序列（LIS）长度为给定值 $k$ 的排列的精确均匀抽样。对于 $k \in Θ(n)$，我们给出了一种直接拒绝抽样器，其在 word-RAM 模型中的期望运行时间为 $O(n\log\log n)$。该抽样器使用扩展的提议空间，该空间由排列及一个指定的递增子序列组成，并仅接受其指定子序列为最左 LIS 的提议。对于任意 $1\le k\le n$，我们给出了一种基于 Robinson--Schensted 对应的精确抽样器。该算法通过行列式恒等式计算精确的完成计数，从而抽样出对应的 Plancherel 条件化形状，然后对该形状的两个一致杨图进行抽样。直接实现的期望运行时间为 $\tilde O(n^4k^5)$。随后，我们展示了通过使用 Hankel 矩矩阵评估行列式预言机，同样的抽样器可以在期望 $\tilde O(n^3k^4)$ 时间内实现。