Prefix palindromic length of the Sierpinski word
share
The prefix palindromic length p_𝐮(n) of an infinite word 𝐮 is the minimal number of concatenated palindromes needed to express the prefix of length n of 𝐮. This function is surprisingly difficult to study; in particular, the conjecture that p_𝐮(n) can be bounded only if 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 𝐬 and give a complete description of p_𝐬(n).
READ FULL TEXT