Burrows–Wheeler transform and palindromic richness

Abstract

The investigation of the extremal case of the Burrows–Wheeler transform leads to study the words w over an ordered alphabet A = { a 1 , a 2 , … , a k } , with a 1 < a 2 < ⋯ < a k , such that b w t ( w ) is of the form a k n k a k − 1 n k − 1 ⋯ a 2 n 2 a 1 n 1 , for some non-negative integers n 1 , n 2 , … , n k . A characterization of these words in the case | A | = 2 has been given in [Sabrina Mantaci, Antonio Restivo, Marinella Sciortino, Burrows-Wheeler transform and Sturmian words, Information Processing Letters 86 (2003) 241–246], where it is proved that they correspond to the powers of conjugates of standard words. The case | A | = 3 has been settled in [Jamie Simpson, Simon J. Puglisi, Words with simple Burrows-Wheeler transforms, Electronic Journal of Combinatorics 15, (2008) article R83], which also contains some partial results for an arbitrary alphabet. In the present paper we show that such words can be described in terms of the notion of “palindromic richness”, recently introduced in [Amy Glen, Jacques Justin, Steve Widmer, Luca Q. Zamboni, Palindromic richness, European Journal of Combinatorics 30 (2) (2009) 510–531]. Our main result indeed states that a word w such that b w t ( w ) has the form a k n k a k − 1 n k − 1 ⋯ a 2 n 2 a 1 n 1 is strongly rich, i.e. the word w 2 contains the maximum number of different palindromic factors.