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Avoiding vincular patterns on alternating words
A word is alternating if either
(when the word is up-down) or (when the word is
down-up). The study of alternating words avoiding classical permutation
patterns was initiated by the authors in~\cite{GKZ}, where, in particular, it
was shown that 123-avoiding up-down words of even length are counted by the
Narayana numbers.
However, not much was understood on the structure of 123-avoiding up-down
words. In this paper, we fill in this gap by introducing the notion of a
cut-pair that allows us to subdivide the set of words in question into
equivalence classes. We provide a combinatorial argument to show that the
number of equivalence classes is given by the Catalan numbers, which induces an
alternative (combinatorial) proof of the corresponding result in~\cite{GKZ}.
Further, we extend the enumerative results in~\cite{GKZ} to the case of
alternating words avoiding a vincular pattern of length 3. We show that it is
sufficient to enumerate up-down words of even length avoiding the consecutive
pattern and up-down words of odd length avoiding the
consecutive pattern to answer all of our enumerative
questions. The former of the two key cases is enumerated by the Stirling
numbers of the second kind.Comment: 25 pages; To appear in Discrete Mathematic
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