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 (), the number of -array words with separations over alphabet [] and show that for any ≥ 0, the growth sequence ( ,()) converges to a characterized limit, independent of . In addition, we study the asymptotic behavior of the random variable , the number of staircase separations in a random word in [] and obtain several limit theorems for the distribution of , including a law of large numbers, a central limit theorem, and the exact growth rate of the entropy of . Finally, we obtain similar results, including growth limits, for longest -staircase subwords and subsequences.