Pattern-avoiding alternating words

TL;DR

This paper studies pattern-avoiding alternating words, providing enumeration formulas, bijections with poset order ideals, and revealing connections with Narayana numbers for specific pattern avoidances.

Contribution

It introduces the enumeration of pattern-avoiding alternating words, establishes bijections with poset order ideals, and connects pattern avoidance counts to Narayana numbers.

Findings

Number of 123-avoiding up-down words of even length equals Narayana numbers.

Bijective proof linking 132-avoiding and 123-avoiding up-down words.

Formulas for counting all other 3-pattern avoidances in alternating words.

Abstract

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.