On the number of balanced words of given length and height over a two letter alphabet
نویسندگان
چکیده
The first investigations on discrete lines are dated back to E.B. Christoffel (Chr75), A. Markoff (Mar82) and more recently to G.A. Hedlund and H. Morse (MH40) who introduced the terminology of sturmian sequences, for the ones defined on a two-letter alphabet and coding lines with irrational slope. These works gave the first theoretical framework for discrete lines. A sequence u ∈ {0, 1}N is sturmian if and only if it is balanced and non-eventually periodic. From the 70’s, H. Freeman (Fre74), A. Rosenfeld (Ros74) and S. Hung (Hun85) extended these investigations to lines with rational slope and studied discrete segments. In (Rev91), J.-P. Reveillès defined arithmetic discrete lines as sets of integer points between two parallel Euclidean lines. More precisely, given a vector v ∈ R and two real numbers μ ∈ R and w ∈ R, the arithmetic discrete line D(v, μ, w) with normal vector v ∈, shift μ and thickness w is the subset of Z defined by D(v, μ, w) = { x ∈ Z, −w2 ≤ 〈v,x〉+ μ < w 2 } , where 〈x,y〉 = x1y1 + x2y2 denotes the usual scalar product in the canonical basis {e1, e2} of R. Moreover, if w = ‖v ‖∞ (resp. w = ‖v ‖1), then the arithmetic discrete line D(v, μ, w) is called naive (resp. standard). There exists a direct relation between naive (resp. standard) discrete arithmetic lines and sturmian sequences. Indeed, given a sturmian sequence u ∈ {0, 1}N, if one associates the letters 0 and 1 with
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عنوان ژورنال:
- Discrete Mathematics & Theoretical Computer Science
دوره 12 شماره
صفحات -
تاریخ انتشار 2010