Circular Super patterns and Zigzag constructions

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.