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

Toufik Mansour, Reza Rastegar, Alexander Roitershtein

Research output: Contribution to journalArticlepeer-review


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

Original languageEnglish
Article number103099
JournalEuropean Journal of Combinatorics
StatePublished - May 2020

Bibliographical note

Publisher Copyright:
© 2020 Elsevier Ltd

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics


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