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arXiv · 1505.04078

Pattern-avoiding alternating words

Emma L. L. Gao·Sergey Kitaev·Philip B. Zhang
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Abstract

A word $w=w_1w_2\cdots w_n$ is alternating if either $w_1 w_3 \cdots$ (when the word is up-down) or $w_1>w_2 w_4<\cdots$ (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.

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Emma L. L. Gao, Sergey Kitaev, Philip B. Zhang. 2015-05-15. Pattern-avoiding alternating words. https://arxiv.org/abs/1505.04078

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