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The Encyclopedia of Integer Sequences
"... This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs ..."
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Cited by 631 (15 self)
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This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs to generate the sequence, references, links to relevant web pages, and other
When can you fold a map
 In Proceedings of the 7th International Workshop on Algorithms and Data Structures
, 2001
"... Abstract. We explore the following problem: given a collection of creases on a piece of paper, each assigned a folding direction of mountain or valley, is there a flat folding by a sequence of simple folds? There are several models of simple folds; the simplest onelayer simple fold rotates a portio ..."
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Cited by 8 (4 self)
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Abstract. We explore the following problem: given a collection of creases on a piece of paper, each assigned a folding direction of mountain or valley, is there a flat folding by a sequence of simple folds? There are several models of simple folds; the simplest onelayer simple fold rotates a portion of paper about a crease in the paper by ¦ � Æ. We first consider the analogous questions in one dimension lower—bending a segment into a flat object—which lead to interesting problems on strings. We develop efficient algorithms for the recognition of simply foldable 1D crease patterns, and reconstruction of a sequence of simple folds. Indeed, we prove that a 1D crease pattern is flatfoldable by any means precisely if it is by a sequence of onelayer simple folds. Next we explore simple foldability in two dimensions, and find a surprising contrast: “map ” folding and variants are polynomial, but slight generalizations are NPcomplete. Specifically, we develop a lineartime algorithm for deciding foldability of an orthogonal crease pattern on a rectangular piece of paper, and prove that it is (weakly) NPcomplete to decide foldability of (1) an orthogonal crease pattern on a orthogonal piece of paper, (2) a crease pattern of axisparallel and diagonal (45degree) creases on a square piece of paper, and (3) crease patterns without a mountain/valley assignment. 1
Counting mountainvalley assignments for flat folds
 Ars Combinatoria
, 2003
"... We develop a combinatorial model of paperfolding for the purposes of enumeration. A planar embedding of a graph is called a crease pattern if it represents the crease lines needed to fold a piece of paper into something. A flat fold is a crease pattern which lies flat when folded, i.e. can be presse ..."
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Cited by 5 (1 self)
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We develop a combinatorial model of paperfolding for the purposes of enumeration. A planar embedding of a graph is called a crease pattern if it represents the crease lines needed to fold a piece of paper into something. A flat fold is a crease pattern which lies flat when folded, i.e. can be pressed in a book without crumpling. Given a crease pattern C =(V,E),amountainvalley (MV) assignment is a function f: E →{M,V} which indicates which crease lines are convex and which are concave, respectively. A MV assignment is valid if it doesn’t force the paper to selfintersect when folded. We examine the problem of counting the number of valid MV assignments for a given crease pattern. In particular we develop recursive functions that count the number of valid MV assignments for flat vertex folds, crease patterns with only one vertex in the interior of the paper. We also provide examples, especially those of Justin, that illustrate the difficulty of the general multivertex case. 1
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 5 (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.
The Combinatorics of Flat Folds: a Survey ∗
"... We survey results on the foldability of flat origami models. The main topics are the question of when a given crease pattern can fold flat, the combinatorics of mountain and valley creases, and counting how many ways a given crease pattern can be folded. In particular, we explore generalizations of ..."
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Cited by 4 (0 self)
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We survey results on the foldability of flat origami models. The main topics are the question of when a given crease pattern can fold flat, the combinatorics of mountain and valley creases, and counting how many ways a given crease pattern can be folded. In particular, we explore generalizations of Maekawa’s and Kawasaki’s Theorems, develop a necessary and sufficient condition for a given assignment of mountains and valleys to fold up in a special case of single vertex folds, and describe recursive formulas to enumerate the number of ways that single vertex in a crease pattern can be folded. 1
A Fast Algorithm To Generate Open Meandric Systems and Meanders
"... An open meandric system is a planar configuration of acyclic curves crossing an infinite horizontal line in the plane such that the curves may extend in both horizontal directions. We present a fast, recursive algorithm to exhaustively generate open meandric systems with n crossings. We then illustr ..."
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Cited by 3 (3 self)
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An open meandric system is a planar configuration of acyclic curves crossing an infinite horizontal line in the plane such that the curves may extend in both horizontal directions. We present a fast, recursive algorithm to exhaustively generate open meandric systems with n crossings. We then illustrate how to modify the algorithm to generate unidirectional open meandric systems (the curves extend only to the right) and nonisomorphic open meandric systems where equivalence is taken under horizontal reflection. Each algorithm can be modified to generate systems with exactly k curves. In the unidirectional case when k=1, we can apply a minor modification along with some additional optimization steps to yield the first fast and simple algorithm to generate open meanders.
ON THE REPRESENTATION OF MEANDERS
"... ABSTRACT. We will introduce a new approach for studying plane meanders. The set of all meanders of order n possesses a natural order structure and forms a graded poset. We will show how these representations can be used to develop an efficient and very flexible construction algorithm and how to obta ..."
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ABSTRACT. We will introduce a new approach for studying plane meanders. The set of all meanders of order n possesses a natural order structure and forms a graded poset. We will show how these representations can be used to develop an efficient and very flexible construction algorithm and how to obtain counting formulae for meanders. 1.
Meanders and Stamp Foldings: Fast Generation Algorithms
, 2008
"... By considering a permutation representation for meanders, semimeanders, and stamp foldings we construct a new data structure that will allow us to extend the order of a given meander, semimeander, or stamp folding in constant time. Then using this data structure, we develop a constant amortized ti ..."
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By considering a permutation representation for meanders, semimeanders, and stamp foldings we construct a new data structure that will allow us to extend the order of a given meander, semimeander, or stamp folding in constant time. Then using this data structure, we develop a constant amortized time algorithm to generate all semimeanders of order n. By maintaining the windfactor for semimeanders and applying an additional optimization, the algorithm can be modified to produce the fastest known algorithm to generate meanders. Finally, by handling an additional special case, the semimeander algorithm can be modified to generate all stamp foldings of order n in constant amortized time.
Stamp Foldings, Semimeanders, and Open Meanders: Fast Generation Algorithms
"... By considering a permutation representation for stampfoldings and semimeanders we construct treelike data structures that will allow us to generate these objects in constant amortized time. Additionally, by maintaining the windfactor and applying an additional optimization, the algorithm for semi ..."
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By considering a permutation representation for stampfoldings and semimeanders we construct treelike data structures that will allow us to generate these objects in constant amortized time. Additionally, by maintaining the windfactor and applying an additional optimization, the algorithm for semimeanders can be
PART A: FOLDING OF REGULAR LATTICES
, 2005
"... Geometrically constrained statistical systems on regular and random lattices: ..."
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Geometrically constrained statistical systems on regular and random lattices: