论文标题
Sierpinski Word的前缀palindromic长度
Prefix palindromic length of the Sierpinski word
论文作者
论文摘要
前缀palindromic长度$ p _ {\ mathbf {u}}（n）$的无限单词$ \ mathbf {u} $是表达长度$ n $ $ n $ $ n $的前缀$ \ mathbf {u} $所需的最小数量。这个功能很难研究。特别是，只有$ p _ {\ mathbf {u}}（n）$的猜想只有在2013年以来最终是周期性开放的，只有在$ \ mathbf {u} $最终开放时才能有限。一个较新的猜想涉及该期间的前缀palindromic的长度，这似乎不是$ 2的，并且如果不是$ 2 $ 2 $ - $ 2的范围，那么它是一个唯一的，并且是一个事实。 $ 2 $自动单词的非规范功能。 但是，对于其他一些$ k $的自动词，众所周知，前缀palindromic长度为$ k $ regular。在这里，我们将这些单词的列表添加到Sierpinski Word $ \ Mathbf {s} $中，并给出$ p _ {\ Mathbf {s}}}（n）$的完整描述。
The prefix palindromic length $p_{\mathbf{u}}(n)$ of an infinite word $\mathbf{u}$ is the minimal number of concatenated palindromes needed to express the prefix of length $n$ of $\mathbf{u}$. This function is surprisingly difficult to study; in particular, the conjecture that $p_{\mathbf{u}}(n)$ can be bounded only if $\mathbf{u}$ is ultimately periodic is open since 2013. A more recent conjecture concerns the prefix palindromic length of the period doubling word: it seems that it is not $2$-regular, and if it is true, this would give a rare if not unique example of a non-regular function of a $2$-automatic word. For some other $k$-automatic words, however, the prefix palindromic length is known to be $k$-regular. Here we add to the list of those words the Sierpinski word $\mathbf{s}$ and give a complete description of $p_{\mathbf{s}}(n)$.
