Pattern-avoiding alternating words

A word w=w1w2⋯wnw=w1w2⋯wn is alternating if either w1w3⋯w1w3⋯ (when the word is up-down) or w1>w2w4<⋯w1>w2w4<⋯ (when the word is down-up). In this paper, we initiate the study of (pattern-avoiding) alternating words. We enumerate up-down (equivalently, down-up) words via finding a bijection with order ideals of a certain poset. Further, we show that the number of 123-avoiding up-down words of even length is given by the Narayana numbers, which is also the case, shown by us bijectively, with 132-avoiding up-down words of even length. We also give formulas for enumerating all other cases of avoidance of a permutation pattern of length 3 on alternating words.