A connection between palindromic and factor complexity using return words

In this paper we prove that for any infinite word w whose set of factors is closed under reversal, the following conditions are equivalent:(I)all complete returns to palindromes are palindromes;(II)P(n)+P(n+1)=C(n+1)−C(n)+2 for all n, where P (resp. C) denotes the palindromic complexity (resp. factor complexity) function of w, which counts the number of distinct palindromic factors (resp. factors) of each length in w.