English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 theprocess we revisit beautiful previous work of Enrique Navarrete, RobertTauraso, David Robbins (to whose memory this article is dedicated), and JohnRiordan. We also present two new proofs of John Riordan's recurrence (from1965) for the sequence enumerating permutations without rising and fallingsuccessions (the $r = 1$, $s = 1$ case of the title in the sense of absolutevalue). The first is fully automatic using the (continuous) Almkvist-Zeilbergeralgorithm, while the second is purely human-generated via an elegantcombinatorial argument. We continue with some open questions and pledgedonations to the OEIS in honor of the solvers. We conclude with a postscriptdescribing interesting ideas of Rintaro Matsuo, that we were made aware ofafter the first version was written, and announce that one of the challengeswas met.

Counting Permutations Where The Difference Between Entries Located $r$ Places Apart Can never be $s$ (For any given positive integers $r$ and $s$)