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

AbstractWe 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