Sharp upper and lower bounds on the length of general Davenport-Schinzel sequences

P. K. Agarwal*, M. Sharir, P. Shor

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

103 Scopus citations


We obtain sharp upper and lower bounds on the maximal length λs(n) of (n, s)-Davenport-Schinzel sequences, i.e., sequences composed of n symbols, having no two adjacent equal elements and containing no alternating subsequence of length s + 2. We show that (i) λ4(n) = Θ(n·2α(n)); (ii) for s > 4, λs(n) ≤ n·2(α(n)) (s - 2) 2 + Cs(n) if s is even and λs(n) ≤ n·2(α(n)) (s - 3) 2log(α(n)) + Cs(n) if s is odd, where Cs(n) is a function of α(n) and s, asymptotically smaller than the main term; and finally (iii) for even values of s > 4, λs(n) = Ω(n·2Ks(α(n)) (s - 2) 2 + Qs(n)), where Ks = (( (s - 2) 2)!)-1 and Qs is a polynomial in α(n) of degree at most (s - 4) 2.

Original languageEnglish
Pages (from-to)228-274
Number of pages47
JournalJournal of Combinatorial Theory. Series A
Issue number2
StatePublished - Nov 1989


FundersFunder number
Digital Equipment Corporation
Israeli National Council for Research and Development
National Science FoundationNSF-DCR-83-20085
Office of Naval ResearchNOOO14-82-K-0381
International Business Machines Corporation
National Center for Research and Development


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