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

Abstract

A generalized Davenport–Schinzel sequence is one over a finite alphabet whose subsequences are not isomorphic to a forbidden subsequence σ. What is the maximum length of such a σ-free sequence, as a function of its alphabet size n? Is the extremal function linear or nonlinear? And what characteristics of σ determine the answers to these questions? It is known that such sequences have length at most n ⋅ 2 ( α ( n ) ) O ( 1 ) , where α is the inverse-Ackermann function and the O ( 1 ) depends on σ. 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 σ ∈ { a , b , c } ⁎ over a three-letter alphabet. Specifically, the only repetition-free minimally nonlinear forbidden sequences are a b a b a and a b c a c b c . 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, Nivaschʼs upper bounds on alternating sequences of the form ( a b ) t and ( a b ) t a , for t ⩾ 3 , can be extended to forbidden sequences of the form ( a a b b ) t and ( a a b b ) t a . 4. Finally, we show that the absence of simple subsequences in σ tells us nothing about σʼs extremal function. For example, for any t, there exists a σ t avoiding a b a b a whose extremal function is Ω ( n ⋅ 2 α 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.