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75
Alice’s Adventures in Wonderland
, 1916
"... "What is the use, " thought Alice, "of a book without pictures and ..."
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"What is the use, " thought Alice, "of a book without pictures and
Numeration systems, linear recurrences, and regular sets
 Inform. and Comput
, 1994
"... A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2,... expresses a nonnegative integer n as a sum n = � i j=0 ajuj. In this case we say the string aiai−1 · · · a1a0 is a representation for n. If gcd(u0, u1,...) = g, then every sufficiently large mult ..."
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A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2,... expresses a nonnegative integer n as a sum n = � i j=0 ajuj. In this case we say the string aiai−1 · · · a1a0 is a representation for n. If gcd(u0, u1,...) = g, then every sufficiently large multiple of g has some representation. If the lexicographic ordering on the representations is the same as the usual ordering of the integers, we say the numeration system is orderpreserving. In particular, if u0 = 1, then the greedy representation, obtained via the greedy algorithm, is orderpreserving. We prove that, subject to some technical assumptions, if the set of all representations in an orderpreserving numeration system is regular, then the sequence u = (uj)j≥0 satisfies a linear recurrence. The converse, however, is not true. The proof uses two lemmas about regular sets that may be of independent interest. The first shows that if L is regular, then the set of lexicographically greatest strings of every length in L is also regular. The second shows that the number of strings of length n in a regular language L is bounded by a constant (independent of n) iff L is the finite union of sets of the form xy ∗ z. 1
A Simple AlphabetIndependent FMIndex
 INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
"... We design a succinct fulltext index based on the idea of Huffmancompressing the text and then applying the BurrowsWheeler transform over it. The resulting structure can be searched as an FMindex, with the benefit of removing the sharp dependence on the alphabet size, σ, present in that structu ..."
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We design a succinct fulltext index based on the idea of Huffmancompressing the text and then applying the BurrowsWheeler transform over it. The resulting structure can be searched as an FMindex, with the benefit of removing the sharp dependence on the alphabet size, σ, present in that structure. On a text of length n with zeroorder entropy H0, our index needs O(n(H0 + 1)) bits of space, without any significant dependence on σ. The average search time for a pattern of length m is O(m(H0 + 1)), under reasonable assumptions. Each position of a text occurrence can be located in worst case time O((H0 + 1)log n), while any text substring of length L can be retrieved in O((H0 + 1)L) average time in addition to the previous worst case time. Our index provides a relevant space/time tradeoff between existing succinct data structures, with the additional interest of being easy to implement. We also explore other coding variants alternative to Huffman and exploit their synchronization properties. Our experimental results on various types of texts show that our indexes are highly competitive in the space/time tradeoff map.
A COUNTING BASED PROOF OF THE GENERALIZED ZECKENDORF’S THEOREM
, 2004
"... We give a counting based short proof of the generalized Zeckendorf’s theorem claiming that every positive integer can be uniquely represented as a sum of generalized Fibonacci numbers of order l with no l consecutive indices. 1. ..."
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We give a counting based short proof of the generalized Zeckendorf’s theorem claiming that every positive integer can be uniquely represented as a sum of generalized Fibonacci numbers of order l with no l consecutive indices. 1.
Extensions and restrictions of Wythoff’s game preserving its P positions
 Journal of Combinatorial Theory, Series A
"... Abstract. We consider extensions and restrictions of Wythoff’s game having exactly the same set of P positions as the original game. No strict subset of rules give the same set of P positions. On the other hand, we characterize all moves that can be adjoined while preserving the original set of P po ..."
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Abstract. We consider extensions and restrictions of Wythoff’s game having exactly the same set of P positions as the original game. No strict subset of rules give the same set of P positions. On the other hand, we characterize all moves that can be adjoined while preserving the original set of P positions. Testing if a move belongs to such an extended set of rules is shown to be doable in polynomial time. Many arguments rely on the infinite Fibonacci word, automatic sequences and the corresponding number system. With these tools, we provide new twodimensional morphisms generating an infinite picture encoding respectively P positions of Wythoff’s game and moves that can be adjoined. 1.
From Fibonacci Numbers to Central Limit Type Theorems
 http://arxiv.org/abs/1008.3202 FROM FIBONACCI NUMBERS TO CENTRAL LIMIT TYPE THEOREMS 51
"... A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of nonconsecutive Fibonacci numbers {Fn}∞n=1. Lekkerkerker proved that the average number of summands for integers in [Fn, Fn+1) is n/(ϕ 2+1), with ϕ the golden mean. This has been generalized to the follow ..."
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A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of nonconsecutive Fibonacci numbers {Fn}∞n=1. Lekkerkerker proved that the average number of summands for integers in [Fn, Fn+1) is n/(ϕ 2+1), with ϕ the golden mean. This has been generalized to the following: given nonnegative integers c1, c2,..., cL with c1, cL> 0 and recursive sequence
Automaticity II: Descriptional Complexity in the Unary Case
 Comput. Sci
, 1995
"... Let \Sigma and \Delta be finite alphabets, and let f be a map from \Sigma to \Delta. Then the deterministic automaticity of f , A f (n), is defined to be the size of the minimum finitestate machine that correctly computes f on all inputs of size n. A similar definition applies to languages L. We ..."
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Let \Sigma and \Delta be finite alphabets, and let f be a map from \Sigma to \Delta. Then the deterministic automaticity of f , A f (n), is defined to be the size of the minimum finitestate machine that correctly computes f on all inputs of size n. A similar definition applies to languages L. We denote the nondeterministic analogue (for languages L) of automaticity by NL (n). In a previous paper, J. Shallit and Y. Breitbart examined the properties of this measure of descriptional complexity in the case j\Sigmaj 2. In this paper, we continue the study of automaticity, focusing on the case where j\Sigmaj = 1. Research supported in part by DMS9206784. y Research supported in part by a grant from NSERC. Partial support under NSF Grant DCR 9208639 and the Wisconsin Alumni Research Foundation. We prove that A f (n) n + 1 \Gamma blog ` nc, where ` = j\Deltaj. We also prove that A f (n) ? n \Gamma 2 log ` n \Gamma 2 log ` log ` n for almost all functions f . In the nondetermi...
THE AVERAGE GAP DISTRIBUTION FOR GENERALIZED ZECKENDORF DECOMPOSITIONS
"... Abstract. An interesting characterization of the Fibonacci numbers is that, if we write them as F1 = 1, F2 = 2, F3 = 3, F4 = 5,..., then every positive integer can be written uniquely as a sum of nonadjacent Fibonacci numbers. This is now known as Zeckendorf’s theorem [21], and similar decompositio ..."
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Abstract. An interesting characterization of the Fibonacci numbers is that, if we write them as F1 = 1, F2 = 2, F3 = 3, F4 = 5,..., then every positive integer can be written uniquely as a sum of nonadjacent Fibonacci numbers. This is now known as Zeckendorf’s theorem [21], and similar decompositions exist for many other sequences {Gn+1 = c1Gn+ · · ·+ cLGn+1−L} arising from recurrence relations. Much more is known. Using continued fraction approaches, Lekkerkerker [15] proved the average number of summands needed for integers in [Gn, Gn+1) is on the order of CLekn for a nonzero constant; this was improved by others to show the number of summands has Gaussian fluctuations about this mean. Koloğlu, Kopp, Miller and Wang [16, 17] recently recast the problem combinatorially, reproving and generalizing these results. We use this new perspective to investigate the distribution of gaps between summands. We explore the average behavior over all m ∈ [Gn, Gn+1) for special choices of the ci’s. Specifically, we study the case where each ci ∈ {0, 1} and there is a g such that there are always exactly g−1 zeros between two nonzero ci’s; note this includes the Fibonacci, Tribonacci and many other important special cases. We prove there are no gaps of length less than g, and the probability of a gap of length j> g decays
On cobweb posets’ most relevant codings
, 2009
"... One considers here acyclic digraphs named KoDAGs (****) which represent the outmost general chains of dibicliques denoting thus the outmost general chains of binary relations. Because of this fact KoDAGs start to become an outstanding concept of nowadays investigation. We propose here examples of ..."
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One considers here acyclic digraphs named KoDAGs (****) which represent the outmost general chains of dibicliques denoting thus the outmost general chains of binary relations. Because of this fact KoDAGs start to become an outstanding concept of nowadays investigation. We propose here examples of codings of KoDAGs looked upon as infinite hyperboxes as well as chains of rectangular hyperboxes in N ∞. Neither of KoDAGs ’ codings considered here is a poset isomorphism with Π = 〈P, ≤〉. Nevertheless every example of coding supplies a new view on possible investigation of KoDAGs properties. The codes proposed here down are by now recognized as most relevant codings for practical purposes including visualization. More than that. Employing quite arbitrary sequences F = {nF}n≥0 infinitely many new representations of natural numbers called an F base or baseF number system representations are introduced. These constitute mixed radixtype numeral systems. F base nonstandard positional numeral systems in which the numerical base varies from position to position have picturesque interpretation due to KoDAGs graphs and their correspondent posets which in turn are endowed on their own with combinatorial interpretation of uniquely assigned to KoDAGs F − nomial coefficients. The baseF number systems are used for KoDAGs ’ coding and are interpreted as chain coordinatization in KoDAGs pictures as well as systems of infinite number of boxes ’ sequences of Fvarying containers capacity of subsequent boxes. Needless to say how crucial is this baseF number system for KoDAGs hence consequently for arbitrary chains of binary relations. New Fbased numeral systems are umbral baseF number systems in a sense to be explained in what follows.
THE JOINT DISTRIBUTION OF GREEDY AND LAZY FIBONACCI EXPANSIONS
, 2002
"... Every nonnegative integer n has at least one digital expansion n = ∑ ɛkFk, k≥2 ..."
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Every nonnegative integer n has at least one digital expansion n = ∑ ɛkFk, k≥2