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 hr,k (n), the number of n-array words with r separations over alphabet [k] and show that for any r ≥ 0, the growth sequence (hr,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.