A Relationship Between Generalized Davenport-Schinzel Sequences and Interval Chains

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 [a0, a1][a1 + 1, a2] . . . [ak−1 + 1, ak] for integers 1 6 a0 6 a1 < . . . < ak 6 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 6 s 6 k 6 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 Ds,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 D5,2d+1(m) = Θ(mαd(m)) for all fixed d > 2. We show that Ds+1,s(m) = (m−d s 2 e b s 2 c ) for all s > 2. We also prove new lower bounds which imply that D5,6(m) = Θ(m log logm) and D5,2d+2(m) = Θ(mαd(m)) for all fixed d > 3.