Sources of Superlinearity in Davenport-Schinzel Sequences

TL;DR

This paper identifies specific forbidden subsequences that cause superlinear growth in the maximum length of Davenport-Schinzel sequences, introducing a new coding method for representing these sequences.

Contribution

It uncovers an infinite family of superlinear forbidden subsequences and presents a novel succinct coding scheme for their representation.

Findings

17 prototypical superlinear forbidden subsequences identified

Some sequences can be extended arbitrarily long with padding

New coding method for representing superlinear subsequences

Abstract

A generalized Davenport-Schinzel sequence is one over a finite alphabet that contains no subsequences isomorphic to a fixed forbidden subsequence. One of the fundamental problems in this area is bounding (asymptotically) the maximum length of such sequences. Following Klazar, let Ex(\sigma,n) be the maximum length of a sequence over an alphabet of size n avoiding subsequences isomorphic to \sigma. It has been proved that for every \sigma, Ex(\sigma,n) is either linear or very close to linear; in particular it is O(n 2^{\alpha(n)^{O(1)}}), where \alpha is the inverse-Ackermann function and O(1) depends on \sigma. However, very little is known about the properties of \sigma that induce superlinearity of \Ex(\sigma,n). In this paper we exhibit an infinite family of independent superlinear forbidden subsequences. To be specific, we show that there are 17 prototypical superlinear forbidden subsequences, some of which can be made arbitrarily long through a simple padding operation. Perhaps the most novel part of our constructions is a new succinct code for representing superlinear forbidden subsequences.