Generalized Davenport-Schinzel sequences and their 0-1 matrix counterparts

• Authors: S. Pettie
• Venue: Journal of Combinatorial Theory Series A
• Year: 2011

A generalized Davenport-Schinzel sequence is one over a finite alphabet whose subsequences are not isomorphic to a forbidden subsequence @s. What is the maximum length of such a @s-free sequence, as a function of its alphabet size n? Is the extremal function linear or nonlinear? And what characteristics of @s determine the answers to these questions? It is known that such sequences have length at most n@?2^(^@a^(^n^)^)^^^O^^^(^^^1^^^), where @a is the inverse-Ackermann function and the O(1) depends on @s. We resolve a number of open problems on the extremal properties of generalized Davenport-Schinzel sequences. Among our results:1.We give a nearly complete characterization of linear and nonlinear @s@?{a,b,c}^@? over a three-letter alphabet. Specifically, the only repetition-free minimally nonlinear forbidden sequences are ababa and abcacbc. 2.We prove there are at least four minimally nonlinear forbidden sequences. 3.We prove that in many cases, doubling a forbidden sequence has no significant effect on its extremal function. For example, Nivasch@?s upper bounds on alternating sequences of the form (ab)^t and (ab)^ta, for t=3, can be extended to forbidden sequences of the form (aabb)^t and (aabb)^ta. 4.Finally, we show that the absence of simple subsequences in @s tells us nothing about @s@?s extremal function. For example, for any t, there exists a @s"t avoiding ababa whose extremal function is @W(n@?2^@a^^^t^(^n^)). Most of our results are obtained by translating questions about generalized Davenport-Schinzel sequences into questions about the density of 0-1 matrices avoiding certain forbidden submatrices. We give new and often tight bounds on the extremal functions of numerous forbidden 0-1 matrices.