2023/11/03 by Kevin Beanland, Hùng Việt Chu, Beanland, Kevin +1 · 1 citation
Mathematics · #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2311.01926
openalex publication_date 2023/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A nonempty set A⊂ℕ is ℓ-strong Schreier if min A\geqslant ℓ|A|-ℓ+1. We define a set of positive integers to be sparse if either the set has at most two numbers or the differences between consecutive numbers in increasing order are non-decreasing. This note establishes a connection between sparse Schreier-type sets and (restricted) partition numbers. One of our results states that if Gn,ℓ consists of partitions of n that contain no parts in \2, …, ℓ\, and An,ℓ := \A⊂ \1, …, n\ : n∈ A, A is sparse and ℓ-strong Schreier\, then |An,ℓ| = |Gn-1,ℓ|, n, ℓ∈ ℕ. The special case Gn-1, 1 consists of all partitions of n-1. Besides partitions, integer compositions are also investigated.