Sequence saturation
Publication Date
1-15-2025
Document Type
Article
Publication Title
Discrete Applied Mathematics
Volume
360
DOI
10.1016/j.dam.2024.09.034
First Page
382
Last Page
393
Abstract
In this paper, we introduce saturation and semisaturation functions of sequences, and we prove a number of fundamental results about these functions. Given a forbidden sequence u with r distinct letters, we say that a sequence s on a given alphabet is u-saturated if s is r-sparse, u-free, and adding any letter from the alphabet to an arbitrary position in s violates r-sparsity or induces a copy of u. We say that s is u-semisaturated if s is r-sparse and adding any letter from the alphabet to s violates r-sparsity or induces a new copy of u. Let the saturation function Sat(u,n) denote the minimum possible length of a u-saturated sequence on an alphabet of size n, and let the semisaturation function Ssat(u,n) denote the minimum possible length of a u-semisaturated sequence on an alphabet of size n. For alternating sequences, we determine both the saturation function and the semisaturation function up to a constant multiplicative factor. We show for every sequence that the semisaturation function is always either O(1) or Θ(n). For the saturation function, we show that every sequence u has either Sat(u,n)≥n or Sat(u,n)=O(1). For every sequence with 2 distinct letters, we show that the saturation function is always either O(1) or Θ(n).
Funding Number
111-2115-M-008-010-MY2
Funding Sponsor
Ministry of Science and Technology, Taiwan
Keywords
Davenport–Schinzel sequence, Extremal combinatorics, Pattern avoidance, Saturation
Department
Mathematics and Statistics
Recommended Citation
Anand, Jesse Geneson, Suchir Kaustav, and Shen Fu Tsai. "Sequence saturation" Discrete Applied Mathematics (2025): 382-393. https://doi.org/10.1016/j.dam.2024.09.034
