  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.
