区间的施赖尔集、超级施赖尔集与卡特兰数
Schreier Sets of Intervals, Super-Schreier Sets, and Catalan Numbers
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中文总结 AI 辅助
该数学研究定义区间型施赖尔集、$k$-超级施赖尔集，证明其计数序列分别满足特定特征多项式对应的递推关系与带多项式余项的斐波那契型递推。
中文摘要 AI 辅助
有限非空集合$F\subset\mathbb{N}$若满足$\min F\ge |F|$则称为施赖尔集。首先，我们证明区间型施赖尔集的计数序列满足线性递推关系并计算初始计数；两个整数区间若其并集非区间则称为分离的，设$\mathcal{J}_{k,n}$为恰好由$k$个分离区间构成的施赖尔集的集合，则序列$(|\mathcal{J}_{k,n}|)_{n=1}^\infty$满足特征多项式$p_k(x)=(x-1)^{2k+1}(x+1)^k$。此外，我们引入$k$-超级施赖尔集的新概念，设$\mathcal{S}_{k,n}$为最大值为$n$的$k$-超级施赖尔集的集合，证明序列$(|\mathcal{S}_{k,n}|)_{n=1}^\infty$满足带余项的斐波那契型递推，余项可表示为$n$的多项式。
英文摘要
A finite nonempty set $F\subset\mathbb{N}$ is Schreier if $\min F\ge |F|$. First, we prove a linear recurrence relation and compute initial counts for Schreier sets consisting of intervals. Two intervals of integers are separated if their union is not an interval. If $\mathcal J_{k,n}$ is the collection of Schreier sets that are the union of exactly $k$ separated intervals, then the sequence $(|\mathcal{J}_{k,n}|)_{n=1}^\infty$ satisfies the characteristic polynomial $p_k(x) = (x-1)^{2k+1}(x+1)^k$. Furthermore, we introduce the new concept of $k$-super-Schreier sets and let $\mathcal{S}_{k,n}$ denote the collection of $k$-super-Schreier sets whose maximum is $n$. We show that the sequence $(|\mathcal{S}_{k,n}|)_{n=1}^\infty$ satisfies a Fibonacci-type recurrence with a remainder term expressible as a polynomial of $n$.
发表机构
- Washington and Lee University（华盛顿与李大学）
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