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Fast Algorithms to Generate Necklaces, Unlabeled Necklaces, and Irreducible Polynomials over GF(2)
, 2000
"... this paper ## Sawada 23 developed an algorithm to generate kary bracelets in constant ## amortized time. Proskurowski et al. 17 show that the orbits of the ' Z. Z. Z . composition of b and d can be generated in amortized Oktime, which is CAT if k is fixed. It remains an interesting challenge ..."
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Cited by 23 (9 self)
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this paper ## Sawada 23 developed an algorithm to generate kary bracelets in constant ## amortized time. Proskurowski et al. 17 show that the orbits of the ' Z. Z. Z . composition of b and d can be generated in amortized Oktime, which is CAT if k is fixed. It remains an interesting challenge to develop efficient algorithms for the other compositions
Compression of periodic complementary sequences and applications, Des. Codes Cryptogr. first published online: 24 July 2013
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A Fast Algorithm for Generating NonIsomorphic Chord Diagrams
 SIAM J. Discrete Math
"... Using a new string representation, we develop two algorithms for generating nonisomorphic chord diagrams. Experimental evidence indicates that the latter of the two algorithms runs in constant amortized time. In addition, we use simple counting techniques to derive a formula for the number of nonis ..."
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Cited by 4 (2 self)
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Using a new string representation, we develop two algorithms for generating nonisomorphic chord diagrams. Experimental evidence indicates that the latter of the two algorithms runs in constant amortized time. In addition, we use simple counting techniques to derive a formula for the number of nonisomorphic chord diagrams. 1.
Generating All Partitions: A Comparison of Two Encodings
, 2009
"... Integer partitions may be encoded as either ascending or descending compositions for the purposes of systematic generation. Many algorithms exist to generate all descending compositions, yet none have previously been published to generate all ascending compositions. We develop three new algorithms t ..."
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Cited by 4 (0 self)
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Integer partitions may be encoded as either ascending or descending compositions for the purposes of systematic generation. Many algorithms exist to generate all descending compositions, yet none have previously been published to generate all ascending compositions. We develop three new algorithms to generate all ascending compositions and compare these with descending composition generators from the literature. We analyse the new algorithms and provide new and more precise analyses for the descending composition generators. In each case, the ascending composition generation algorithm is substantially more efficient than its descending composition counterpart. We develop a new formula for the partition function p(n) as part of our analysis of the lexicographic succession rule for ascending compositions. 1
Computationally Efficient Recursions for TopOrder Invariant Polynomials with Applications ∗
"... evaluation of toporder invariant polynomials and moments of ratio of quadratic forms in normal random variables. ” Hillier first became involved as a referee of that earlier paper. His contribution has been confined mainly to suggesting the generating function approach, simplifying some of the proo ..."
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Cited by 2 (1 self)
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evaluation of toporder invariant polynomials and moments of ratio of quadratic forms in normal random variables. ” Hillier first became involved as a referee of that earlier paper. His contribution has been confined mainly to suggesting the generating function approach, simplifying some of the proofs, and contributing a few additional results. We are grateful to Plamen Koev, Peter Phillips, Serge Provost, Marko Riedel and two anonymous referees for helpful comments and suggestions. Kan gratefully acknowledges financial support from the National Bank Financial of Canada.
Musical Scales, Integer Partitions, Necklaces, and Polygons
"... A musical scale can be viewed as a subsequence of notes taken from a chromatic sequence. Given integers (N, K) N> K we use particular integer partitions of N into K parts to construct distinguished scales. We show that a natural geometric realization of these scales results in maximal polygons. 1 ..."
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A musical scale can be viewed as a subsequence of notes taken from a chromatic sequence. Given integers (N, K) N> K we use particular integer partitions of N into K parts to construct distinguished scales. We show that a natural geometric realization of these scales results in maximal polygons. 1
Generating Bracelets with Fixed Content
, 2011
"... We present an algorithm to generate bracelets with fixed content. An analysis shows that the algorithm runs in constant amortized time. The algorithm can be applied to efficiently list all nonisomorphic unicyclic graphs with n vertices. ..."
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We present an algorithm to generate bracelets with fixed content. An analysis shows that the algorithm runs in constant amortized time. The algorithm can be applied to efficiently list all nonisomorphic unicyclic graphs with n vertices.
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"... Computationally efficient recursions for toporder invariant polynomials with applications ..."
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Computationally efficient recursions for toporder invariant polynomials with applications
Binary Quadratic Forms as Dessins
, 2012
"... We show that the class of every primitive indefinite binary quadratic form is naturally represented by an infinite graph (named çark) with a unique cycle embedded on a conformal annulus. This cycle is called the spine of the çark. Every çark is an infinite dessin and determines an annular uniform ..."
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We show that the class of every primitive indefinite binary quadratic form is naturally represented by an infinite graph (named çark) with a unique cycle embedded on a conformal annulus. This cycle is called the spine of the çark. Every çark is an infinite dessin and determines an annular uniformization of the modular curve. Every choice of a base edge of a fixed çark specifies an indefinite binary quadratic form in the class represented by the çark. The proper automorphism group of a form is identified with the fundamental group of its çark. Reduced forms in the class represented by a çark correspond to some distinguished base edges of its spine. The reduction algorithm of Gauss is the process of moving the base edge in the direction of the spine of the çark. Ambiguous and reciprocal classes are represented by çarks with symmetries. Periodic çarks represent classes of nonprimitive forms. 1 Introduction. The Euclidean algorithm is the process of comparison of commensurable magni