关于$k$ -阿贝尔等价类的基数
On cardinalities of $k$-abelian equivalence classes
摘要：
两个词$u$和$v$是$k$ -阿贝尔等价的，如果对于长度最多为$k$的每个词$x$，$x$作为因子在$u$和$v$中出现的次数相等。$k$ -阿贝尔等价概念是介于阿贝尔等价和词的相等之间的一个中间概念。本文研究了由$k$ -阿贝尔等价导出的等价类，主要研究了等价类的基数。特别是，我们感兴趣的是单$k$ -阿贝尔类的数量，即，类只包含一个元素。我们发现之间的连接的单例类和循环分解的de Bruijn图。我们表明，包含一个单一的元素的长度$n$的字的类的数量是为了$\mathcal O(n^{N_m(k-1)-1})$，其中$N_m(l) = \tfrac{1}{l}\sum_{d\mid l} φ(d)m^{l/d}$是在$m$ -ary字母表的长度$l$项链的数量。我们猜想上界是尖锐的。我们还注意到，$k$甚至和$m = 2$，下界$Ω(n^{N_m(k-1)-1})$如下从一个老猜想的存在格雷码的项链长度为奇数。我们验证了这个猜想的项链长度达15。
英文摘要：
Two words $u$ and $v$ are $k$-abelian equivalent if, for each word $x$ of length at most $k$, $x$ occurs equally many times as a factor in both $u$ and $v$. The notion of $k$-abelian equivalence is an intermediate notion between the abelian equivalence and the equality of words. In this paper, we study the equivalence classes induced by the $k$-abelian equivalence, mainly focusing on the cardinalities of the classes. In particular, we are interested in the number of singleton $k$-abelian classes, i.e., classes containing only one element. We find a connection between the singleton classes and cycle decompositions of the de Bruijn graph. We show that the number of classes of words of length $n$ containing one single element is of order $\mathcal O(n^{N_m(k-1)-1})$, where $N_m(l) = \tfrac{1}{l}\sum_{d\mid l} φ(d)m^{l/d}$ is the number of necklaces of length $l$ over an $m$-ary alphabet. We conjecture that the upper bound is sharp. We also remark that, for $k$ even and $m = 2$, the lower bound $Ω(n^{N_m(k-1)-1})$ follows from an old conjecture on the existence of Gray codes for necklaces of odd length. We verify this conjecture for necklaces of length up to 15.
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