态射族和词族的二项式复杂性刻画
Characterizations of families of morphisms and words via binomial complexities
摘要：
如果长度最多为$k$的每个子字在两个字中出现的次数相同，则两个字是$k$二项式等价的。无限字的$k$ -二项式复杂度是一个计数函数，它将$n$映射到由其长度因子$n$表示的$k$ -二项式等价类的数量。Cassaigne等人[Int. J. Found. Comput.美国，22（4）（2011）]描述了一个态射族，我们称之为Parikh-共线的，作为那些将所有单词映射到具有有界$1$ -二项式复杂度的单词的态射。首先，我们扩展了这一特征：他们映射有界$k$ -二项式复杂度的字有界$(k+1)$ -二项式复杂度。因此，Parikh-共线态射的不动点被证明具有有界的$k$ -二项式复杂性的所有$k$。其次，我们给出了一个新的Sturmian词的$k$ -二项式复杂度的刻画。然后，我们的特点是经常性的话，对于一些$k$，相同的$j$ -二项式复杂度的Thue-Morse字的所有$j\le k$。最后，受到Lejeune提出的问题的启发，我们研究了$k$ -和$(k+1)$ -无穷词的二项式复杂度之间的关系，以及与通常的因子复杂度的联系。
英文摘要：
Two words are $k$-binomially equivalent if each subword of length at most $k$ occurs the same number of times in both words. The $k$-binomial complexity of an infinite word is a counting function that maps $n$ to the number of $k$-binomial equivalence classes represented by its factors of length $n$. Cassaigne et al. [Int. J. Found. Comput. S., 22(4) (2011)] characterized a family of morphisms, which we call Parikh-collinear, as those morphisms that map all words to words with bounded $1$-binomial complexity. Firstly, we extend this characterization: they map words with bounded $k$-binomial complexity to words with bounded $(k+1)$-binomial complexity. As a consequence, fixed points of Parikh-collinear morphisms are shown to have bounded $k$-binomial complexity for all $k$. Secondly, we give a new characterization of Sturmian words with respect to their $k$-binomial complexity. Then we characterize recurrent words having, for some $k$, the same $j$-binomial complexity as the Thue-Morse word for all $j\le k$. Finally, inspired by questions raised by Lejeune, we study the relationships between the $k$- and $(k+1)$-binomial complexities of infinite words; as well as the link with the usual factor complexity.
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