2025/09/05 by Hùng Việt Chu, Chu, Hung Viet, Julian A. King +8
Mathematics · #05A15 (secondary) #05A19 (primary) #11B37 #11Y55 #Combinatorics (math.CO) #FOS: Mathematics #Functional Equations Stability Results #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2509.05158
openalex publication_date 2025/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A nonempty set F is Schreier if min F≥ |F|. Bird observed that counting Schreier sets in a certain way produces the Fibonacci sequence. Since then, various connections between variants of Schreier sets and well-known sequences have been discovered. Building on these works, we prove a linear recurrence for the sequence that counts multisets F with min F≥ p|F|. In particular, if we let A(s)p, n := \F⊂ \\underbrace1, …, 1s, …, \underbracen-1, …, n-1s, n\ : n∈ F and min F≥ p|F|\, then |A(s)p, n| = ∑i=0s|A(s)p, n-1-ip|. If we color s copies of the same integer by different colors from 1 to s, i.e., B(s)p, n:= \F⊂ \11, …, 1s, …, (n-1)1, …, (n-1)s, n\ : n∈ F and min F≥ p|F|\, then |B(s)p, n| = ∑i=0s \binomsi| B(s)p, n-1-ip|. Lastly, we count Schreier sets that do not admit multiples of a given integer u≥ 2 and witness linear recurrences whose coefficients are drawn from the uth row of the Pascal triangle and have alternating signs, except possibly the last one.