Weighted Schreier-type Sets and the Fibonacci Sequence

2024/05/23 by Hùng Việt Chu, Chu, Hung Viet, Zachary Louis Vasseur +1
Physics and Astronomy · #11B37 #11Y55 #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2405.19352

openalex publication_date 2024/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a finite set A⊂ℕ and k∈ ℕ, let ω_k(A) = ∑_{i∈ A, i≠ k}1. For each n∈ ℕ, define a_{k, n} = |\E⊂ ℕ : E = ∅ or ω_k(E) lt; min E\leqslant max E\leqslant n\|. First, we prove that a_k,k+ℓ = 2F_k+ℓ, for all ℓ\geqslant 0 and k\geqslant ℓ+2, where F_n is the nth Fibonacci number. Second, we show that |\E⊂ ℕ : max E = n+1, min E gt; ω_2,3(E), and |E|≠ 2\| = F_n, where ω_2,3(E) = ∑_{i∈ E, i≠ 2, 3}1.

Related

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• Schreier Multisets and the s-step Fibonacci Sequences
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