论文标题
计算Schreier套装的工会
Counting Unions of Schreier Sets
论文作者
论文摘要
如果是非空的，则是一个正整数$ f $的子集是Schreier集合，而$ | f | \ leqslant \ min f $（以下$ | f | $是$ f $的基数）。对于每个正整数$ k $，我们将$ k \ mathcal {s} $定义为最多$ k $ schreier套件的所有工会的集合。另外，对于每个正整数$ n $，令$（k \ mathcal {s}）^n $是$ k \ mathcal {s} $的所有集合，最大元素等于$ n $。众所周知，序列$（|（1 \ Mathcal {s}）^n |）_ {n = 1}^\ infty $是fibbonacci序列。特别是，该序列满足线性复发。我们概括了此语句，即，我们表明序列$（|（k \ Mathcal {s}）^n |）_ {n = 1}^\ infty $满足每个正$ k $的线性复发。
A subset of positive integers $F$ is a Schreier set if it is non-empty and $|F|\leqslant \min F$ (here $|F|$ is the cardinality of $F$). For each positive integer $k$, we define $k\mathcal{S}$ as the collection of all the unions of at most $k$ Schreier sets. Also, for each positive integer $n$, let $(k\mathcal{S})^n$ be the collection of all sets in $k\mathcal{S}$ with the maximum element equal to $n$. It is well-known that the sequence $(|(1\mathcal{S})^n|)_{n=1}^\infty$ is the Fibbonacci sequence. In particular, the sequence satisfies a linear recurrence. We generalize this statement, namely, we show that the sequence $(|(k\mathcal{S})^n|)_{n=1}^\infty$ satisfies a linear recurrence for every positive $k$.
