Increasing consecutive patterns in words

@article{Yang2019IncreasingCP,
  title={Increasing consecutive patterns in words},
  author={Mingjia Yang and Doron Zeilberger},
  journal={Journal of Algebraic Combinatorics},
  year={2019},
  volume={51},
  pages={89-101}
}
We show how to enumerate words in $$1^{m_1} \ldots n^{m_n}$$ 1 m 1 … n m n that avoid the increasing consecutive pattern $$12 \ldots r$$ 12 … r for any $$r \ge 2$$ r ≥ 2 . Our approach yields an $$O(n^{s+1})$$ O ( n s + 1 ) algorithm to enumerate words in $$1^s \ldots n^s$$ 1 s … n s , avoiding the consecutive pattern $$1\ldots r$$ 1 … 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… 
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