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

TL;DR

This paper develops methods to count permutations with restrictions on the difference between entries r places apart, revisiting classical results, providing new proofs, and exploring open questions in permutation enumeration.

Contribution

It introduces semi-efficient enumeration techniques for these permutations, revisits and provides new proofs for Riordan's recurrence, and discusses open problems and new ideas in permutation theory.

Findings

Revisited classical permutation enumeration results.

Provided two new proofs of Riordan's recurrence.

Extended understanding of permutations with difference constraints.

Abstract

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.