论文标题
斐波那契序列和schreier-type集的注释
A Note on the Fibonacci Sequence and Schreier-type Sets
论文作者
论文摘要
如果$ a = \ emptyset $或$ \ min a \ ge | a | $，则据说$ a $ a $ a $ a的正整数。我们提供一张徒地图，以证明序列$（| \ Mathcal {k} _ {n，p，q} |）_ {n = 1}^\ infty $（对于固定$ p \ ge 1 $和$ q \ ge 2 $） \ {1，\ ldots，n \} \，：\，\ mbox {er as} a = \ emptySet \ mbox \ mbox {或}（\ max a- \ max a- \ max_2 a = p \ mbox {and} \ mbox {and} \ min a \ ge | $ | a | \ ge 2 $。当$ p = 1 $和$ q = 2 $时，我们有$（| \ Mathcal {k} _ {n，1，2} |）_ {n = 1}^\ infty $是fibonacci序列。作为推论，我们获得了序列$（f_n + n）_ {n = 1}^\ infty $的新组合解释。
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 $(F_n + n)_{n=1}^\infty$.
