Palindromic Length of Words with Many Periodic Palindromes
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The palindromic length PL(v) of a finite word v is the minimal number of palindromes whose concatenation is equal to v. In 2013, Frid, Puzynina, and Zamboni conjectured that: If w is an infinite word and k is an integer such that PL(u)≤ k for every factor u of w then w is ultimately periodic. Suppose that w is an infinite word and k is an integer such PL(u)≤ k for every factor u of w. Let Ω(w,k) be the set of all factors u of w that have more than √(k^-1| u|) palindromic prefixes. We show that Ω(w,k) is an infinite set and we show that for each positive integer j there are palindromes a,b and a word u∈Ω(w,k) such that (ab)^j is a factor of u and b is nonempty. Note that (ab)^j is a periodic word and (ab)^ia is a palindrome for each i≤ j. These results justify the following question: What is the palindromic length of a concatenation of a suffix of b and a periodic word (ab)^j with "many" periodic palindromes? It is known that |PL(uv)-PL(u)|≤PL(v), where u and v are nonempty words. The main result of our article shows that if a,b are palindromes, b is nonempty, u is a nonempty suffix of b, | ab| is the minimal period of aba, and j is a positive integer with j≥3PL(u) then PL(u(ab)^j)-PL(u)≥ 0.
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