Increasing Consecutive Patterns in Words
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.
 Publication:

arXiv eprints
 Pub Date:
 May 2018
 arXiv:
 arXiv:1805.06077
 Bibcode:
 2018arXiv180506077Y
 Keywords:

 Mathematics  Combinatorics;
 05A05;
 05A15
 EPrint:
 After the first version of the current paper was posted, Justin Troyka pointed out that our Theorem 1 is not new and it goes back to Ira Gessel. Thus we have this second version