TY - JOUR

T1 - Sharp upper and lower bounds on the length of general Davenport-Schinzel sequences

AU - Agarwal, P. K.

AU - Sharir, M.

AU - Shor, P.

N1 - Funding Information:
*Work on this paper by the first two authors has been supported by Office of Naval Research Grant NOOO14-82-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.

PY - 1989/11

Y1 - 1989/11

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=2842562910&partnerID=8YFLogxK

U2 - 10.1016/0097-3165(89)90032-0

DO - 10.1016/0097-3165(89)90032-0

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AN - SCOPUS:2842562910

SN - 0097-3165

VL - 52

SP - 228

EP - 274

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

IS - 2

ER -