论文标题
计算排列，其中位于$ r $ place的条目之间的差异永远不会是$ s $（对于任何给定的正整数$ r $和$ s $）
Counting Permutations Where The Difference Between Entries Located $r$ Places Apart Can never be $s$ (For any given positive integers $r$ and $s$)
论文作者
论文摘要
鉴于正整数$ r $和$ s $，我们使用包容性排斥，瓷砖加权计数和动态编程，以列举标题中提到的排列类别。在此过程中，我们重新审视了Enrique Navarrete，Robert Tauraso，David Robbins（本文的记忆）和John Riordan的精美作品。我们还提供了约翰·里奥丹（John Riordan）复发的两个新证据（从1965年起），列举了列出的排列，而没有上升和下降继承（$ r = 1 $，$ s = 1 $ case of标题的标题案例，从绝对价值看）。第一个是使用（连续的）Almkvist-Zeilberger算法完全自动的，而第二个则是通过优雅的组合论证纯粹的人类生成的。为了纪念求解者，我们继续向OEI提出一些开放的问题，并向OEIS捐款。最后，我们以一篇文章描述了Rintaro Matsuo的有趣思想，在写了第一个版本后，我们就意识到了，并宣布满足了其中一个挑战。
Given positive integers $r$ and $s$, we use inclusion-exclusion, weighted-counting of tilings, and dynamical programming, in order to enumerate, semi-efficiently, the classes of permutations mentioned in the title. In the process we revisit beautiful previous work of Enrique Navarrete, Robert Tauraso, David Robbins (to whose memory this article is dedicated), and John Riordan. We also present two new proofs of John Riordan's recurrence (from 1965) for the sequence enumerating permutations without rising and falling successions (the $r = 1$, $s = 1$ case of the title in the sense of absolute value). The first is fully automatic using the (continuous) Almkvist-Zeilberger algorithm, while the second is purely human-generated via an elegant combinatorial argument. We continue with some open questions and pledge donations to the OEIS in honor of the solvers. We conclude with a postscript describing interesting ideas of Rintaro Matsuo, that we were made aware of after the first version was written, and announce that one of the challenges was met.
