Computing the number of cubic runs in standard Sturmian words

The standard Sturmian   words (standard words in short) are extensively studied in combinatorics of words. They are complicated enough to have many interesting properties, and, at the same time, due to their recurrent structure, they are highly compressible. In this paper, we present compact formulas for the number of cubic runs in any standard word  ww (denoted by ρ(3)(w)ρ(3)(w)). We show also that lim sup|w|→∞ρ(3)(w)|w|=3Φ+29Φ+4≈0.36924841, where Φ=5+12 is the golden ratio, and present a sequence of strictly growing standard words achieving this limit. The exact asymptotic ratio here is irrational, contrary to the situation of squares and runs in the same class of words. Furthermore, we design an efficient algorithm for computing the number of cubic runs in standard words in linear time with respect to the size of a directive sequence, i.e., the compressed representation describing the word (recurrences). The explicit size of a word can be exponential with respect to this representation, and hence this is yet another example of a very fast computation on highly compressible texts.