Complementing and exactly covering sequences
TL;DR: A method of reduction is developed which, given a complementing system of m sequences, leads under certain conditions to a derived complementingSystem of m − 1 sequences.
About: This article is published in Journal of Combinatorial Theory, Series A.The article was published on 1973-01-01 and is currently open access. It has received 71 citations till now. The article focuses on the topics: Prime (order theory) & Integer.
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TL;DR: This paper presents the first unified approach to the study of a new class of finite and infinite words that are 'rich' in palindromes in the utmost sense and proves that a certain class of almost rich words obeys Fraenkel's conjecture.
Abstract: In this paper, we study combinatorial and structural properties of a new class of finite and infinite words that are 'rich' in palindromes in the utmost sense. A characteristic property of the so-called rich words is that all complete returns to any palindromic factor are themselves palindromes. These words encompass the well-known episturmian words, originally introduced by the second author together with Droubay and Pirillo in 2001 [X. Droubay, J. Justin, G. Pirillo, Episturmian words and some constructions of de Luca and Rauzy, Theoret. Comput. Sci. 255 (2001) 539-553]. Other examples of rich words have appeared in many different contexts. Here we present the first unified approach to the study of this intriguing family of words. Amongst our main results, we give an explicit description of the periodic rich infinite words and show that the recurrent balanced rich infinite words coincide with the balanced episturmian words. We also consider two wider classes of infinite words, namely weakly rich words and almost rich words (both strictly contain all rich words, but neither one is contained in the other). In particular, we classify all recurrent balanced weakly rich words. As a consequence, we show that any such word on at least three letters is necessarily episturmian; hence weakly rich words obey Fraenkel's conjecture. Likewise, we prove that a certain class of almost rich words obeys Fraenkel's conjecture by showing that the recurrent balanced ones are episturmian or contain at least two distinct letters with the same frequency. Lastly, we study the action of morphisms on (almost) rich words with particular interest in morphisms that preserve (almost) richness. Such morphisms belong to the class of P-morphisms that was introduced by Hof, Knill, and Simon in 1995 [A. Hof, O. Knill, B. Simon, Singular continuous spectrum for palindromic Schrodinger operators, Comm. Math. Phys. 174 (1995) 149-159].
162 citations
Cites background from "Complementing and exactly covering ..."
...2 Connection to Fraenkel’s conjecture Fraenkel’s conjecture [17] is a problem concerning balance that arose in a number-theoretic context and has remained unsolved for over thirty years....
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TL;DR: The rich theory of infinite episturmian words which generalize to any finite alphabet is surveyed, in a rather resembling way, the well-known family of Sturmian Words on two letters.
Abstract: In this paper, we survey the rich theory of infinite episturmian words which generalize to any finite alphabet, in a rather resembling way, the well-known family of Sturmian words on two letters. After recalling definitions and basic properties, we consider episturmian morphisms that allow for a deeper study of these words. Some properties of factors are described, including factor complexity, palindromes, fractional powers, frequencies, and return words. We also consider lexicographical properties of episturmian words, as well as their connection to the balance property, and related notions such as finite episturmian words, Arnoux-Rauzy sequences, and "episkew words" that generalize the skew words of Morse and Hedlund.
103 citations
Cites result from "Complementing and exactly covering ..."
...The importance of the above result lies in the fact that it supports Fraenkel’s conjecture [56]: a problem that arose in a number-theoretic context and has remained unsolved for over thirty years....
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TL;DR: In this article, the characteristic function of the infinite sequence [n θ] itself was generated from the finite sequence [ n θ]-[nθ] by using shift operators.
Abstract: Let θ = θ(k) be the positive root of θ 2 + (k-2)θ-k = 0. Let f(n) = [(n + l)θ]-[nθ] for positive integers n, where [x] denotes the greatest integer in x. Then the elements of the infinite sequence (f(l), f(2), f(3),…) can be rapidly generated from the finite sequence (f(l), f(2),…,f(k)) by means of certain shift operators. For k = 1 we can generate (the characteristic function of) the sequence [n θ] itself in this manner.
100 citations
TL;DR: In this article, it was shown that the optimal deterministic routing in stochastic event graphs is such a sequence, where each letter is distributed as "evenly" as possible and appears with a given rate.
Abstract: The objective pursued in this paper is two-fold. The first part addresses the following combinatorial problem: is it possible to construct an infinite sequence over n letters where each letter is distributed as “evenly” as possible and appears with a given rate? The second objective of the paper is to use this construction in the framework of optimal routing in queuing networks. We show under rather general assumptions that the optimal deterministic routing in stochastic event graphs is such a sequence.
99 citations
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193 citations
TL;DR: In this article, it was shown that if α and β are positive irrational numbers satisfying 1 then the sets [nα], [nβ], n = 1, 2, …, are complementary with respect to the set of all positive integers, if and only if β and α are irrational.
Abstract: The following result is well known (as usual, [x]denotes the integral part of x): (A) Let α and β be positive irrational numbers satisfying 1 Then the sets [nα], [nβ], n= 1, 2, …, are complementary with respect to the set of all positive integers]see, e.g. (1; 2; 4; 5; 6; 7; 8; 10; 13; 14; 15; 16). In some of these references the result, or a special case thereof, is mentioned in connection with Wythoff's game, with or without proof. It appears that Beatty (4) was the originator of the problem. The theorem has a converse, and the following holds: (B) Let α and β be positive. The sets [nα] and [nβ], n = 1, 2, …, are complementary with respect to the set of all positive integers if and only if α and β are irrational, and (1) holds.
79 citations
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TL;DR: In this article, the authors present a problem arising out of the theory of a certain game, which they call Problem Arising from the Theory of a Certain Game (PASG).
Abstract: (1927). On a Problem Arising out of the Theory of a Certain Game. The American Mathematical Monthly: Vol. 34, No. 10, pp. 516-521.
35 citations