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Fertility monotonicity and average complexity of the stack-sorting map
European Journal of Combinatorics ( IF 1.0 ) Pub Date : 2020-12-10 , DOI: 10.1016/j.ejc.2020.103276
Colin Defant

Let Dn denote the average number of iterations of West’s stack-sorting map s that are needed to sort a permutation in Sn into the identity permutation 123n. We prove that 0.62433λlim infnDnnlim supnDnn35(78log2)0.87289, where λ 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 |s1(σ)||s1(s(σ))| for all σSn, where equality holds if and only if σ=123n. 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 Sn.



中文翻译:

肥力单调性和堆栈分类图的平均复杂度

dñ 表示West的堆栈排序图的平均迭代次数 s 在排序中需要的 小号ñ 进入身份置换 123ñ。我们证明062433λlim infñdññlim supñdññ357-8日志2087289 哪里 λ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|s-1个σ||s-1个sσ| 对所有人 σ小号ñ,当且仅当等式成立时 σ=123ñ。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小号ñ

更新日期:2020-12-10
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