2022/05/27 by Hùng Việt Chu, Chu, Hung Viet
Physics and Astronomy · Mathematics · Biochemistry, Genetics and Molecular Biology · #Advanced Mathematical Theories and Applications #Advanced Combinatorial Mathematics #Fractal and DNA sequence analysis
paper · pdf · doi:10.48550/arxiv.2205.14260
A set A of positive integers is said to be Schreier if either A = ∅ or min A≥ |A|. We give a bijective map to prove the recurrence of the sequence (|Kn, p, q|)n=1^∞ (for fixed p≥ 1 and q≥ 2), where Kn, p, q = \A⊂ \1, …, n\ : either A = ∅ or (max A-max2 A = p and min A≥ |A|≥ q)\ and max2 A is the second largest integer in A, given that |A|≥ 2. When p = 1 and q=2, we have that (|Kn, 1, 2|)n=1^∞ is the Fibonacci sequence. As a corollary, we obtain a new combinatorial interpretation for the sequence (Fn + n)n=1^∞.