The Number of Runs in Sturmian Words

Abstract

Abstract. Denote by S the class of standard Sturmian words. It is a class of highly compressible words extensively studied in combinatorics of words, including the well known Fibonacci words. The suffix automata for these words have a very particular structure. This implies a simple characterization (described in the paper by the Structural Lemma) of the periods of runs (maximal repetitions) in Sturmian words. Using this characterization we derive an explicit formula for the number ρ(w) of runs in words w ∈ S, with respect to their recurrences (directive sequences). We show thatfor each w ∈ S, and there is an infinite sequence of strictly growing words w k ∈ S such that lim k→∞. The complete understanding of the function ρ for a large class S of complicated words is a step towards better understanding of the structure of runs in words. We also show how to compute the number of runs in a standard Sturmian word in linear time with respect to the size of its compressed representation (recurrences describing the word). This is an example of a very fast computation on texts given implicitly in terms of a special grammar-based compressed representation (usually of logarithmic size with respect to the explicit text).

Computing the number of cubic runs in standard Sturmian words

“…Finally, we have computed the asymptotic limit of the ratio of the number of cubic runs to the length of the word. In contrast to the maximal repetitions ratio (0.8; see [2]) and the distinct squares ratio (0.9; see [3]), in this case the limit is an irrational number:…”

“…We reduce the problem of counting large cubic runs to that of counting medium cubic runs, using morphic representation of standard words introduced in Section 3. We use here similar approach as in [2]. Let h be a morphism, let v = a 1 a 2 .…”

“…For the class S of standard Sturmian words, there are known exact formulas for the number of runs and squares and their asymptotic behavior; see [2,22] for details. In this case, we have lim n→∞ ρ(n)…”

The maximal number of cubic runs in a word

“…The best known result runs(n) 1.029n is due to Crochemore et al [6], but it is conjectured [11] that runs(n) < n. The maximal number of runs was also studied for special types of words and tight bounds were established for Fibonacci words [11,18] and more generally Sturmian words [1]. The combinatorial analysis of runs is strongly related to the problem of estimation of the maximal number of squares in a word.…”

“…Example binary words for which the maximal number of cubic runs is attained are shown in the following Table 2. n 3 4 5 6 7 8 9 1 0 1 1 cubic-runs 2 (n) 1 1 1 2 2 2 3 3 3 n 12 13 14 15 16 17 18 19 20 cubic-runs 2 (n) 4 [10]. The best known result runs(n) 1.029n is due to Crochemore et al [6], but it is conjectured [11] that runs(n) < n. The maximal number of runs was also studied for special types of words and tight bounds were established for Fibonacci words [11,18] and more generally Sturmian words [1].…”