The factor composition matrix of sequences

AbstractLet S={a,b} be a two-letter alphabet and s a sequence over S. An infinite matrix (ti,j)i,j⩾0 is associated with s in the following way: ti,j=1 if s has a factor which contains i times a and j times b; otherwise ti,j=0. This matrix will be called the factor composition matrix (FCM) of the sequence s. In this paper, combinatorial properties of certain sequences are studied via their FCMs. In particular(i)the FCM of the Thue–Morse sequence is shown to be pentadiagonal, and substitutions whose fixed points have the same FCM as the Thue–Morse sequence are determined;(ii)an algorithm for computing the FCM of a sturmian sequence is presented;(iii)the FCMs of fixed points of invertible substitutions are characterized in terms of their singular decompositions