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A word $w=w_1w_2\cdots w_n$ is alternating if either $w_1<w_2>w_3<w_4>\cdots$(when the word is up-down) or $w_1>w_2<w_3>w_4<\cdots$ (when the word isdown-up). The study of alternating words avoiding classical permutationpatterns was initiated by the authors in~\cite{GKZ}, where, in particular, itwas shown that 123-avoiding up-down words of even length are counted by theNarayana numbers.  However, not much was understood on the structure of 123-avoiding up-downwords. In this paper, we fill in this gap by introducing the notion of acut-pair that allows us to subdivide the set of words in question intoequivalence classes. We provide a combinatorial argument to show that thenumber of equivalence classes is given by the Catalan numbers, which induces analternative (combinatorial) proof of the corresponding result in~\cite{GKZ}.  Further, we extend the enumerative results in~\cite{GKZ} to the case ofalternating words avoiding a vincular pattern of length 3. We show that it issufficient to enumerate up-down words of even length avoiding the consecutivepattern $\underline{132}$ and up-down words of odd length avoiding theconsecutive pattern $\underline{312}$ to answer all of our enumerativequestions. The former of the two key cases is enumerated by the Stirlingnumbers of the second kind.

Avoiding vincular patterns on alternating words