# Staircase patterns in words: subsequences, subwords, and separation number

## TL;DR

This paper investigates the combinatorial properties and asymptotic behaviors of staircase patterns in words, including subsequences, subwords, and separation numbers, providing exact and limit results for these structures.

## Contribution

It introduces new asymptotic and exact results for staircase subsequences, subwords, and separation numbers, extending understanding of their growth and distribution in words.

## Key findings

- Growth rate of $h_{r,k}(n)$ converges to a limit independent of $r$
- Distribution of staircase separations follows a law of large numbers and a central limit theorem
- Exact entropy growth rate of staircase separation in random words

## Abstract

We revisit staircases for words and prove several exact as well as asymptotic results for longest left-most staircase subsequences and subwords and staircase separation number, the latter being defined as the number of consecutive maximal staircase subwords packed in a word. We study asymptotic properties of the sequence $h_{r,k}(n),$ the number of $n$-array words with $r$ separations over alphabet $[k]$ and show that for any $r\geq 0,$ the growth sequence $\big(h_{r,k}(n)\big)^{1/n}$ converges to a characterized limit, independent of $r.$ In addition, we study the asymptotic behavior of the random variable $\mathcal{S}_k(n),$ the number of staircase separations in a random word in $[k]^n$ and obtain several limit theorems for the distribution of $\mathcal{S}_k(n),$ including a law of large numbers, a central limit theorem, and the exact growth rate of the entropy of $\mathcal{S}_k(n).$ Finally, we obtain similar results, including growth limits, for longest $L$-staircase subwords and subsequences.