2026/08/06 by Hung Viet Chu, Mariam Khaduri, Moiz M. Khokhar +1 · 1 voice
Mathematics · #math.CO #msc:11B37 #msc:11B39 #msc:11B50 #msc:11B65
22 pages, 4 tables
arxiv created 2026/08/06 · arxiv updated 2026/08/07
A finite nonempty set F⊂ℕ is Schreier if min F≥ |F|. First, we prove a linear recurrence relation and compute initial counts for Schreier sets consisting of intervals. Two intervals of integers are separated if their union is not an interval. If \mathcal Jk,n is the collection of Schreier sets that are the union of exactly k separated intervals, then the sequence (|Jk,n|)n=1^∞ satisfies the characteristic polynomial pk(x) = (x-1)2k+1(x+1)k. Furthermore, we introduce the new concept of k-super-Schreier sets and let Sk,n denote the collection of k-super Schreier sets whose maximum is n. We show that the sequence (|Sk,n|)n=1^∞ satisfies a Fibonacci-type recurrence with a remainder term expressible as a polynomial of n.