Increasing Consecutive Patterns in Words

TL;DR

This paper presents an efficient algorithm for enumerating words avoiding specific consecutive increasing patterns, enabling the extension of OEIS sequences and the creation of new ones, with implementations in Maple.

Contribution

It introduces a novel $O(n^{s+1})$ algorithm for pattern avoidance in words, applicable to any pattern length and word size, and provides computational tools.

Findings

Extended OEIS sequences with new terms

Developed an $O(n^{s+1})$ enumeration algorithm

Provided Maple packages for implementation

Abstract

We show how to enumerate words in $1^{m_1} \dots n^{m_n}$ that avoid the increasing consecutive pattern $12 \dots r$ for any $r \geq 2$. Our approach yields an $O(n^{s+1})$ algorithm to enumerate words in $1^s \dots n^s$, avoiding the consecutive pattern $1\dots r$, for any $s$, and any $r$. This enables us to supply many more terms to quite a few OEIS sequences, and create new ones. We also treat the more general case of counting words with a specified number of the pattern of interest (the avoiding case corresponding to zero appearances). This article is accompanied by three Maple packages implementing our algorithms.