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

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 $r\ge 0,$ the growth sequence $(h_{r,k}(n){)}^{1/n}$ 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.