Avoiding vincular patterns on alternating words

TL;DR

This paper studies alternating words avoiding certain permutation patterns, introduces the concept of cut-pairs to classify these words, and extends enumeration results to vincular pattern avoidance, revealing connections to Catalan and Stirling numbers.

Contribution

It introduces cut-pairs for classifying 123-avoiding alternating words and extends enumeration to vincular pattern avoidance, providing new combinatorial proofs and results.

Findings

Number of equivalence classes equals Catalan numbers.

Enumeration of certain alternating words relates to Stirling numbers.

Extended results for vincular pattern avoidance 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). The study of alternating words avoiding classical permutation patterns was initiated by the authors in~\cite{GKZ}, where, in particular, it was shown that 123-avoiding up-down words of even length are counted by the Narayana numbers. However, not much was understood on the structure of 123-avoiding up-down words. In this paper, we fill in this gap by introducing the notion of a cut-pair that allows us to subdivide the set of words in question into equivalence classes. We provide a combinatorial argument to show that the number of equivalence classes is given by the Catalan numbers, which induces an alternative (combinatorial) proof of the corresponding result in~\cite{GKZ}. Further, we extend the enumerative results in~\cite{GKZ} 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 $\underline{132}$ and up-down words of odd length avoiding the consecutive pattern $\underline{312}$ to answer all of our enumerative questions. The former of the two key cases is enumerated by the Stirling numbers of the second kind.