Total palindrome complexity of finite words

The palindrome complexity function palwpalw of a word ww attaches to each n∈Nn∈N the number of palindromes (factors equal to their mirror images) of length nn contained in ww. The number of all the nonempty palindromes in a finite word is called the total palindrome complexity of that word. We present exact bounds for the total palindrome complexity and construct words which have any palindrome complexity between these bounds, for binary alphabets as well as for alphabets with the cardinal greater than 2. Denoting by Mq(n)Mq(n) the average number of palindromes in all words of length nn over an alphabet with qq letters, we present an upper bound for Mq(n)Mq(n) and prove that the limit of Mq(n)/nMq(n)/n is 0. A more elaborate estimation leads to Mq(n)=O(n).