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Circular Super patterns and Zigzag constructions

Hariprasad Manjunath, Raisa Dsouza

TL;DR

The paper defines circular $k$-superpatterns and proves a general upper bound $L_{ ext{circ}}(k) \le L(k-1) + 1$ by reducing to linear $(k-1)$-superpatterns. It adapts the Engen–Vatter zigzag framework to the circular setting, introducing a circular score function $S^c$ and a parity-based local cost to analyze pattern embeddings, with parity-driven guarantees for odd $k$. The authors establish exact containment results within zigzag words and present an explicit odd-$k$ construction via breaking ties on a modified zigzag word, supported by computational examples (e.g., $k=5$). They also provide a proof outline and lemmas (distant inverse-descent, layered permutations) to justify the odd-$k construction and discuss open questions on tight bounds and algorithmic generation of minimal circular superpatterns.

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.

Circular Super patterns and Zigzag constructions

TL;DR

The paper defines circular -superpatterns and proves a general upper bound by reducing to linear -superpatterns. It adapts the Engen–Vatter zigzag framework to the circular setting, introducing a circular score function and a parity-based local cost to analyze pattern embeddings, with parity-driven guarantees for odd . The authors establish exact containment results within zigzag words and present an explicit odd- construction via breaking ties on a modified zigzag word, supported by computational examples (e.g., ). They also provide a proof outline and lemmas (distant inverse-descent, layered permutations) to justify the odd-$k construction and discuss open questions on tight bounds and algorithmic generation of minimal circular superpatterns.

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.
Paper Structure (11 sections, 13 theorems, 38 equations, 3 figures, 1 table)

This paper contains 11 sections, 13 theorems, 38 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background and Notations
  3. Circular superpatterns.
  4. Zigzag patterns.
  5. Circular $k$-superpatterns using linear $k-1$ superpattern
  6. Exact subsequence containment in Zigzag words
  7. Parity and local cost.
  8. Circular Score function
  9. Even $k$ case:
  10. A construction for odd circular superpattern
  11. An odd circular superpattern

Key Result

Theorem 3.1

Let $k \ge 2$. If $\pi = (\pi_1, \pi_2, \dots, \pi_L)$ is a linear $(k-1)$-superpattern, then the permutation is a circular $k$-superpattern: every cyclic equivalence class of permutations of length $k$ contains a representative that appears as an order-isomorphic subsequence of $\gamma$. Consequently,

Figures (3)

  • Figure 1: $zz(4,4)$
  • Figure 2: $zz(5,5)$
  • Figure 3: $zz(3,3)$ and corresponding $\zeta$

Theorems & Definitions (40)

  • Example 2.1
  • Definition 2.2: Rotation of a permutation
  • Definition 2.3: Circular containment of a pattern
  • Example 2.4
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Example 3.3
  • Lemma 4.1
  • proof
  • ...and 30 more