广义Schreier集的高阶Fibonacci序列
Higher Order Fibonacci Sequences from Generalized Schreier sets
摘要：
Schreier集合$S$是具有$\min S\ge |S|$的自然数的子集。已知序列$(a_{1,n})$，其中$$a_{1,n}\ :=\ |\{S\subseteq \mathbb{N}\,:\,\max S = n\mbox{ and } \min S \ge |S|\}|,$$是斐波那契序列。推广这一结果，我们证明了对所有的$p\in \mathbb{N}$，序列$(a_{p,n})$，其中$$a_{p, n} \ :=\ |\{S\subseteq \mathbb{N}\,:\,\max S = n\mbox{ and } \min S\ge p|S|\}|,$$有一个高阶线性递归关系。我们通过要求${\rm min}_2 S\ge q |S|$进一步研究，其中$\min_2 S$是$S$的第二小元素。本文证明了序列$(a_{p, q, n})$的一个线性递归关系，其中$$a_{p, q, n} \ :=\ |\{S\subseteq \mathbb{N}\,:\,\max S = n, \min S \ge p|S|\mbox{ and } {\rm min}_2 S\ge q|S|\}|,$$和讨论了$(a_{q, n})$和$(a_{p, q, n})$之间的一个奇怪的关系.
英文摘要：
A Schreier set $S$ is a subset of the natural numbers with $\min S\ge |S|$. It has been known that the sequence $(a_{1,n})$, where $$a_{1,n}\ :=\ |\{S\subseteq \mathbb{N}\,:\,\max S = n\mbox{ and } \min S \ge |S|\}|,$$ is the Fibonacci sequence. Generalizing this result, we prove that for all $p\in \mathbb{N}$, the sequence $(a_{p,n})$, where $$a_{p, n} \ :=\ |\{S\subseteq \mathbb{N}\,:\,\max S = n\mbox{ and } \min S\ge p|S|\}|,$$ has a linear recurrence relation of higher order. We investigate further by requiring that ${\rm min}_2 S\ge q |S|$, where $\min_2 S$ is the second smallest element of $S$. We prove a linear recurrence relation for the sequence $(a_{p, q, n})$, where $$a_{p, q, n} \ :=\ |\{S\subseteq \mathbb{N}\,:\,\max S = n, \min S \ge p|S|\mbox{ and } {\rm min}_2 S\ge q|S|\}|,$$ and discuss a curious relationship between $(a_{q, n})$ and $(a_{p, q, n})$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
