A map-theoretic approach to Davenport-Schinzel sequences

Abstract

An (n, d) Davenport-Schinzel Sequence (more briefly, a DS sequence) is a sequence of symbols selected from 1, 2, …, n, with the properties that (1) no two adjacent symbols are identical, (2) no subsequence of the form abab… has length greater than d, (3) no symbol can be added to the end of the sequence, without violating (1) or (2). It is shown that the set of (n, 3) DS sequences is in one-to-one correspondence with the set of rooted planar maps on n vertices in which every edge of the map is incident with the root face. The number of such sequences and the number of such sequences of longest possible length 2n - 1 is explicitly determined. © 1972, Pacific Journal of Mathematics.