A relationship between generalized Davenport-Schinzel sequences and interval chains

Abstract

Let an (r,s)-formation be a concatenation of s permutations of r distinct letters, and let a block of a sequence be a subsequence of consecutive distinct letters. A k-chain on [1,m] is a sequence of k consecutive, disjoint, nonempty intervals of the form [a[subscript 0],a[subscript 1]][a[subscript 1] + 1,a[subscript 2]]…[a[subscript k−1] + 1,a[subscript k]] for integers 1 ≤ a[subscript 0] ≤ a[subscript 1] <…< a[subscript k] ≤ m, and an s-tuple is a set of s distinct integers. An s-tuple stabs an interval chain if each element of the s-tuple is in a different interval of the chain. Alon et al. (2008) observed similarities between bounds for interval chains and Davenport-Schinzel sequences, but did not identify the cause. We show for all r ≥ 1 and 1 ≤ s ≤ k ≤ m that the maximum number of distinct letters in any sequence S on m + 1 blocks avoiding every (r,s + 1)-formation such that every letter in S occurs at least k + 1 times is the same as the maximum size of a collection X of (not necessarily distinct) k-chains on [1,m] so that there do not exist r elements of X all stabbed by the same s-tuple. Let D[subscript s,k](m) be the maximum number of distinct letters in any sequence which can be partitioned into m blocks, has at least k occurrences of every letter, and has no subsequence forming an alternation of length s. Nivasch (2010) proved that D[subscript 5,2d+1](m) = Θ(mα[subscript d](m)) for all fixed d ≥ 2. We show that D[subscript s+1,s](m) = ([m - [s/2] over [s/2]]) for all s ≥ 2. We also prove new lower bounds which imply that D[subscript 5,6](m) = Θ(mloglogm) and D[subscript 5,2d+2](m) = Θ(mαd(m)) for all fixed d ≥ 3.