A Note on the Fibonacci Sequence and Schreier-type Sets

Abstract: A set $A$ of positive integers is said to be Schreier if either $A = \emptyset$ or $\min A\ge |A|$. We give a bijective map to prove the recurrence of the sequence $(|\mathcal{K}{n, p, q}|){n=1}^\infty$ (for fixed $p\ge 1$ and $q\ge 2$), where $$\mathcal{K}{n, p, q} \ = \ {A\subset {1, \ldots, n}\,:\, \mbox{either }A = \emptyset \mbox{ or } (\max A-\max_2 A = p\mbox{ and }\min A\ge |A|\ge q)}$$ and $\max_2 A$ is the second largest integer in $A$, given that $|A|\ge 2$. When $p = 1$ and $q=2$, we have that $(|\mathcal{K}{n, 1, 2}|){n=1}^\infty$ is the Fibonacci sequence. As a corollary, we obtain a new combinatorial interpretation for the sequence $(F_n + n){n=1}^\infty$.