Forbidden Patterns and the Alternating Derangement Sequence

TL;DR

This paper explores a sequence counting arrangements avoiding specific patterns, revealing its close relationship to derangement numbers and providing new expressions linking these combinatorial objects.

Contribution

It introduces a novel sequence related to pattern-avoiding arrangements and establishes its near-equivalence to derangement numbers, including explicit formulas.

Findings

Sequence Dn counts arrangements avoiding consecutive patterns.

Odd terms of Dn exceed derangements by one.

Even terms of Dn are one less than derangements.

Abstract

In this note we count linear arrangements that avoid certain patterns and show their connection to the derangement numbers. We discuss the sequence Dn, which counts linear arrangements that avoid patterns 12, 23, ..., (n-1)n, n1, and show that this sequence almost follows the derangement sequence itself since the number of its odd terms is one more than the derangement numbers while the number of its even terms is one less. We also express the derangement numbers in terms of these and other arrangements.