Circular Super patterns and Zigzag constructions

TL;DR

This paper introduces circular k-superpatterns, explores their construction from linear superpatterns, and adapts zigzag frameworks to analyze their properties, providing bounds and computational evidence for small cases.

Contribution

It presents the concept of circular superpatterns, a construction method from linear superpatterns, and adapts zigzag analysis to the circular setting, offering new bounds and insights.

Findings

Upper bounds on the length of circular superpatterns

A simplified score function for circular patterns

Computational evidence supporting zigzag constructions for small k

Abstract

In this article, we introduce the notion of circular k-superpatterns, defined as permutations that contain all length-k patterns up to rotation equivalence. We present a construction of a circular superpattern from a linear (k-1)-superpattern and explicitly derive an upper bound on its length. Motivated by the zigzag framework of Engen and Vatter, we adapt and simplify their score function to the circular setting and analyze its parity properties. For odd k, we propose a candidate zigzag construction for circular superpatterns, supported by computational evidence for small values of k.