Increasing Consecutive Patterns in Words

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 $r\ge 2$. 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.