In combinatorics on words, the well-studied factor complexity function $\rho_{\bf x}$ of a sequence ${\bf x}$ over a finite alphabet counts, for any nonnegative integer $n$, the number of distinct length-$n$ factors of $\mathbf{x}$. In this paper, we introduce the reflection complexity function $r_{\bf x}$ to enumerate the factors occurring in a sequence ${\bf x}$, up to reversing the order of symbols in a word. We introduce and prove general results on $r_{\bf x}$ regarding its growth properties and relationship with other complexity functions. We prove a Morse-Hedlund-type result characterizing eventually periodic sequences in terms of their reflection complexity, and we deduce a characterization of Sturmian sequences. Furthermore, we investigate the reflection complexity of quasi-Sturmian, episturmian, $(s+1)$-dimensional billiard, and complementation-symmetric Rote, and rich sequences. Furthermore, we prove that if ${\bf x}$ is $k$-automatic, then $r_{\bf x}$ is computably $k$-regular, and we use the software $\mathtt{Walnut}$ to evaluate the reflection complexity of automatic sequences, such as the Thue-Morse sequence. We note that there are still many unanswered questions about this measure.

翻译：在字组合学中，已得到充分研究的因子复杂度函数 $\rho_{\bf x}$ 用于统计有限字母表上序列 ${\bf x}$ 的相异性：对于任意非负整数 $n$，该函数计算序列 $\mathbf{x}$ 中所有长度为 $n$ 的不同因子的数量。本文引入反射复杂度函数 $r_{\bf x}$，用于枚举序列 ${\bf x}$ 中出现的因子，这些因子在计数时允许单词中符号顺序的逆转。我们提出并证明了关于 $r_{\bf x}$ 的一般性结果，涉及其增长特性以及与其他复杂度函数的关系。我们证明了一个莫尔斯-赫德隆德型结果，该结果通过反射复杂度刻画了最终周期序列，并由此推导出对斯特姆序列的刻画。此外，我们研究了准斯特姆序列、外斯特姆序列、$(s+1)$ 维台球序列、补对称罗特序列以及富序列的反射复杂度。我们还证明，如果 ${\bf x}$ 是 $k$-自动序列，那么 $r_{\bf x}$ 是可计算 $k$-正则的，并利用软件 $\mathtt{Walnut}$ 评估了自动序列（如图埃-莫尔斯序列）的反射复杂度。我们注意到，关于这一度量仍存在许多未解决的问题。