  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 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.
