Sequences of formation width $4$ and alternation length $5$

TL;DR

This paper investigates the formation width of sequence patterns, especially those containing 'ababa', establishing that their extremal function grows proportionally to n times the inverse Ackermann function, thus advancing understanding in combinatorial sequence avoidance.

Contribution

The paper characterizes all sequences with formation width 4 containing 'ababa', linking their extremal function to the inverse Ackermann function, and identifying new classes of sequences with known growth behavior.

Findings

Sequences with fw=4 containing 'ababa' have extremal function Θ(n α(n))

Identified all sequences with formation width 4 containing 'ababa'

Extended the understanding of sequence avoidance extremal functions

Abstract

Sequence pattern avoidance is a central topic in combinatorics. A sequence $s$ contains a sequence $u$ if some subsequence of $s$ can be changed into $u$ by a one-to-one renaming of its letters. If $s$ does not contain $u$, then $s$ avoids $u$. A widely studied extremal function related to pattern avoidance is $Ex(u, n)$, the maximum length of an $n$-letter sequence that avoids $u$ and has every $r$ consecutive letters pairwise distinct, where $r$ is the number of distinct letters in $u$. We bound $Ex(u, n)$ using the formation width function, $fw(u)$, which is the minimum $s$ for which there exists $r$ such that any concatenation of $s$ permutations, each on the same $r$ letters, contains $u$. In particular, we identify every sequence $u$ such that $fw(u)=4$ and $u$ contains $ababa$. The significance of this result lies in its implication that, for every such sequence $u$, we have $Ex(u, n) = \Theta(n \alpha(n))$, where $\alpha(n)$ denotes the incredibly slow-growing inverse Ackermann function. We have thus identified the extremal function of many infinite classes of previously unidentified sequences.