Abstract

We obtain sharp upper and lower bounds on the maximal length λ_s(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·2^α(n)); (ii) for s > 4, λ_s(n) n·2^{(α(n))(s − 2)/2 + C_s(n)} if s is even and λ_s(n) n·2^{(α(n))(s − 3)/2log(α(n)) + C_s(n)} if s is odd, where C_s(n) is a function of α(n) and s, asymptotically smaller than the main term; and finally (iii) for even values of s > 4, λ_s(n) = Ω(n·2^{K_s(α(n))(s − 2)/2 + Q_s(n)}), where K_s = (((s − 2)/2)!)⁻¹ and Q_s is a polynomial in α(n) of degree at most (s − 4)/2.

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*1 Work on this paper by the first two authors has been supported by Office of Naval Research Grant N00014-82-K-0381, by National Science Foundation Grant NSF-DCR-83-20085, and by grants from the Digital Equipment Corporation, and the IBM Corporation. Work by the second author has also been supported by a research grant from the NCRD —the Israeli National Council for Research and Development.