Avoiding vincular patterns on alternating words

Further, we extend the enumerative results in Gao et al. [6] 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 132¯ and up-down words of odd length avoiding the consecutive pattern 312¯ to answer all of our enumerative questions. The former of the two key cases is enumerated by the Stirling numbers of the second kind.