Circular Super patterns and Zigzag constructions

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.

💡 Research Summary

The paper introduces the concept of circular k‑superpatterns, which are permutations written on a circle and required to contain every length‑k permutation up to rotation. After recalling the classical problem of determining the smallest length L(k) of a linear superpattern, the authors define circular containment: a permutation π ∈ S_n circularly contains a pattern σ ∈ S_k if some rotation of σ appears as an order‑isomorphic subsequence of π. The minimal possible length of a circular superpattern is denoted L_circ(k).

The first main contribution is a very simple construction that turns any linear (k − 1)‑superpattern into a circular k‑superpattern. Given a linear (k − 1)‑superpattern π = (π₁,…,π_L), prepend the new maximum element L + 1 to obtain γ = (L + 1, π₁,…,π_L). The authors prove that for every σ ∈ S_k there exists a rotation whose first entry is k; the remaining k − 1 entries are found inside π because π already contains all (k − 1)‑patterns. Consequently γ circularly contains a representative of each rotation class, establishing the bound
 L_circ(k) ≤ L(k − 1) + 1.
Using the Engen–Vatter bound L(k − 1) ≤ ⌈(k − 1)²/2⌉ + ½, this yields the explicit inequality
 L_circ(k) ≤ ⌈k²/2⌉ − k + 2.
The authors verify that the bound is tight for k = 4 (minimal length 6) and close to optimal for k = 5 (a length‑9 circular superpattern exists, while the bound gives 10).

The second major part of the paper adapts the “zigzag” construction of Engen and Vatter to the circular setting. For a fixed alphabet width q, the infinite zigzag word is built from runs R_j that list numbers of matching parity in increasing order for odd j and decreasing order for even j. The finite word zz(m,q) is the concatenation of the first m runs. The authors introduce a parity‑based local cost function C_{x,y} that measures how many additional runs are needed to place y after x in a zigzag word. The formula
 C_{x,y} = δ_{x,y} −