Schreier Sets of Intervals, Super-Schreier Sets, and Catalan Numbers

@misc{chu2026schreier,
  title = {{Schreier Sets of Intervals, Super-Schreier Sets, and Catalan Numbers}},
  author = {Hung Viet Chu and Mariam Khaduri and Moiz M. Khokhar and Ruoan Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.06076},
  eprint = {2608.06076},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.06076},
}

TY  - UNPB
TI  - Schreier Sets of Intervals, Super-Schreier Sets, and Catalan Numbers
AU  - Hung Viet Chu
AU  - Mariam Khaduri
AU  - Moiz M. Khokhar
AU  - Ruoan Zhou
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2608.06076
UR  - https://arxiv.org/abs/2608.06076
AB  - 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 𝒥_k,n is the collection of Schreier sets that are the union of exactly k separated intervals, then the sequence (|𝒥_k,n|)_n=1^∞ satisfies the characteristic polynomial p_k(x) = (x−1)^2k+1(x+1)^k. Furthermore, we introduce the new concept of k-super-Schreier sets and let 𝒮_k,n denote the collection of k-super-Schreier sets whose maximum is n. We show that the sequence (|𝒮_k,n|)_n=1^∞ satisfies a Fibonacci-type recurrence with a remainder term expressible as a polynomial of n.
ER  -