On the number of episturmian palindromes

Abstract

Episturmian words are a suitable generalization to arbitrary alphabets of Sturmian words. In this paper we are interested in the problem of enumerating the palindromes in all episturmian words over a k -letter alphabet A k . We give a formula for the map g k giving for any n the number of all palindromes of length n in all episturmian words over A k . This formula extends to k > 2 a similar result obtained for k = 2 by the second and third authors in 2006. The map g k is expressed in terms of the map P k counting for each n the palindromic prefixes of all standard episturmian words (epicentral words). For any n ≥ 0 , P 2 ( n ) = φ ( n + 2 ) , where φ is the totient Euler function. The map P k plays an essential role also in the enumeration formula for the map λ k counting for each n the finite episturmian words over A k . Similarly to Euler’s function, the behavior of P k is quite irregular. The first values of P k and of the related maps g k , and λ k for 3 ≤ k ≤ 6 have been calculated and reported in the paper. Some properties of P k are shown. In particular, broad upper and lower bounds for P k , as well as for ∑ m = 0 n P k ( m ) and g k , are determined. Finally, some conjectures concerning the map P k are formulated.