AI 中文总结
本研究提出多项式时间算法计算集合[n]的可分错位排列数bₙ，基于生成函数技术，证明bₙ与大Schröder数指数增长常数相同，利用前3000项推测渐近行为，部分回答Vatter的相关问题。
AI 中文摘要
我们给出了一种多项式时间算法，用于计算集合[n]的可分错位排列的数量bₙ。该算法基于生成函数技术，该技术跟踪排列及其占据的对角线，其中每个排列针对每条此类对角线被计数一次。我们给出了可分排列中错位排列所占比例的界，证明了bₙ和大Schröder数具有相同的指数增长常数(3 + 2√2)，并利用前3000项推测更明确的渐近行为。这部分回答了Vatter最近提出的关于可分错位排列的几个问题。
英文摘要
We give a polynomial-time algorithm to compute the number $b_n$ of separable derangements of $[n]$. This algorithm is based on a generating function technique which tracks permutations along with their occupied diagonals, where each permutation is counted once for every such diagonal. We provide bounds for the proportion of separable permutations which are derangements, show that $b_n$ and the large Schröder numbers have the same exponential growth constant $(3 + 2 \sqrt{2})$, and use the first 3000 terms to conjecture more explicit asymptotic behavior. This partially answers several questions about separable derangements recently posed by Vatter.
