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

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 is 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≥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 S_k(n), the number of staircase separations in a random word in [k]ⁿ and obtain several limit theorems for the distribution of S_k(n), including a law of large numbers, a central limit theorem, and the exact growth rate of the entropy of S_k(n). Finally, we obtain similar results, including growth limits, for longest L-staircase subwords and subsequences.