Let ${\mathcal{D}}_n$ denote the average number of iterations of West’s stack-sorting map $s$ that are needed to sort a permutation in $S_n$ into the identity permutation $123\cdots n$. We prove that $0.62433\approx \lambda \le \underset{n\to \infty }{lim\; inf}\frac{{\mathcal{D}}_n}{n}\le \underset{n\to \infty }{lim\; sup}\frac{{\mathcal{D}}_n}{n}\le \frac{3}{5}(7-8log2)\approx 0.87289,$ where $\lambda$ is the Golomb–Dickman constant. Our lower bound improves upon West’s lower bound of 0.23, and our upper bound is the first improvement upon the trivial upper bound of 1. We then show that fertilities of permutations increase monotonically upon iterations of $s$. More precisely, we prove that $\left|s^{-1}\left(\sigma \right)\right|\le \left|s^{-1}\left(s\left(\sigma \right)\right)\right|$ for all $\sigma \in S_n$, where equality holds if and only if $\sigma =123\cdots n$. This is the first theorem that manifests a law-of-diminishing-returns philosophy for the stack-sorting map that Bóna has proposed. Along the way, we note some connections between the stack-sorting map and the right and left weak orders on $S_n$.

让 ${\mathcal{d}}_{ñ}$ 表示West的堆栈排序图的平均迭代次数 $s$ 在排序中需要的 ${\mathrm{小号}}_{ñ}$ 进入身份置换 $123\cdots ñ$。我们证明$0。62433\approx \lambda \le \underset{ñ\to \infty }{lim\; inf}\frac{{\mathcal{d}}_{ñ}}{ñ}\le \underset{ñ\to \infty }{lim\; sup}\frac{{\mathcal{d}}_{ñ}}{ñ}\le \frac{3}{5}（7-8日志2）\approx 0。87289，$ 哪里 $\lambda$是Golomb–Dickman常数。我们的下界在West的下界0.23上得到了改进，而我们的上限是在琐碎的上界1在上的第一个改进。$s$。更确切地说，我们证明$\left|s^{-\mathrm{1个}}（\sigma ）\right|\le \left|s^{-\mathrm{1个}}（s（\sigma ））\right|$ 对所有人 $\sigma \in {\mathrm{小号}}_{ñ}$，当且仅当等式成立时 $\sigma =123\cdots ñ$。这是第一个定理，该定理体现了Bona提出的堆叠排序图的递减收益法则。在此过程中，我们注意到堆栈排序图与左右弱指令之间的一些联系${\mathrm{小号}}_{ñ}$。