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Products of Universal Cycles
"... Universal cycles are generalizations of de Bruijn cycles to combinatorial patterns other than binary strings. We show how to construct a product cycle of two universal cycles, where the window widths of the two cycles may be different. Applications to card tricks are suggested. ..."
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Universal cycles are generalizations of de Bruijn cycles to combinatorial patterns other than binary strings. We show how to construct a product cycle of two universal cycles, where the window widths of the two cycles may be different. Applications to card tricks are suggested.
Universal cycles for permutation classes
"... Abstract. We define a universal cycle for a class of npermutations as a cyclic word in which each element of the class occurs exactly once as an nfactor. We give a general result for cyclically closed classes, and then survey the situation when the class is defined as the avoidance class of a set ..."
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Abstract. We define a universal cycle for a class of npermutations as a cyclic word in which each element of the class occurs exactly once as an nfactor. We give a general result for cyclically closed classes, and then survey the situation when the class is defined as the avoidance class of a set of permutations of length 3, or of a set of permutations of mixed lengths 3 and 4. Résumé. Nous définissons un cycle universel pour une classe de npermutations comme un mot cyclique dans lequel chaque élément de la classe apparaît une unique fois comme nfacteur. Nous donnons un resultat général pour les classes cycliquement closes, et détaillons la situation où la classe de permutations est définie par motifs exclus, avec des motifs de taille 3, ou bien à la fois des motifs de taille 3 et de taille 4.
Research Summary and Research Plan for Garth Isaak
"... this paper. A paper on separation number of a graph [12] arose when a colleague (Jim Driscoll) walked into my office an said `we are working on using wavelets in medical imaging and encountered this combinatorics problem in our work'. Looking at the separation number in a graph is `equivalent&a ..."
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this paper. A paper on separation number of a graph [12] arose when a colleague (Jim Driscoll) walked into my office an said `we are working on using wavelets in medical imaging and encountered this combinatorics problem in our work'. Looking at the separation number in a graph is `equivalent' to looking for powers of Hamiltonian paths in the graph's complement. This has led to my work in [16] and [17]. Working intensively with a Lehigh graduate student (Darren Narayan) this summer on finishing a project he started during a REU (research experiences for undergraduates) is producing a paper (also with Anthony Evans) on representations of graphs modulo n. This problem arose from work of Erdos and Evans on orthogonal latin square graphs [26] [28]. I expect that in the future some of my research will continue to arise from unexpected sources such as these