Further results on staircase (cyclic) words

TL;DR

This paper derives a two-variable generating function for counting variations in staircase cyclic words, refining previous results on staircase words with bounded variation between consecutive letters.

Contribution

It extends prior work by providing generating functions for cyclic staircase words, a new class with bounded variations in a cyclic context.

Findings

Derived the two-variable generating function for cyclic staircase words.

Extended previous results to include cyclic words.

Refined enumeration techniques for words with bounded variations.

Abstract

We find the two-variables generating function for the statistic which counts the number of variations in a word bounded by $1$. Thus, we refine and extend previous results concerning staircase words, which are words in which the variation between all consecutive letters is bounded by $1$. We obtain the analogue results for cyclic words.