  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 $(F_n + n)_{n=1}^\infty$.
