























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.
此内容由惯性聚合(RSS阅读器)自动聚合整理,仅供阅读参考。 原文来自 — 版权归原作者所有。