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Integer partitions, tilings of 2 D -gons and lattices

Matthieu Latapy (2002)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

In this paper, we study two kinds of combinatorial objects, generalized integer partitions and tilings of 2 D -gons (hexagons, octagons, decagons, etc.). We show that the sets of partitions, ordered with a simple dynamics, have the distributive lattice structure. Likewise, we show that the set of tilings of a 2 D -gon is the disjoint union of distributive lattices which we describe. We also discuss the special case of linear integer partitions, for which other dynamical models exist.

Integer Partitions, Tilings of 2D-gons and Lattices

Matthieu Latapy (2010)

RAIRO - Theoretical Informatics and Applications

In this paper, we study two kinds of combinatorial objects, generalized integer partitions and tilings of 2D-gons (hexagons, octagons, decagons, etc.). We show that the sets of partitions, ordered with a simple dynamics, have the distributive lattice structure. Likewise, we show that the set of tilings of a 2D-gon is the disjoint union of distributive lattices which we describe. We also discuss the special case of linear integer partitions, for which other dynamical models exist.

Linear programming duality and morphisms

Winfried Hochstättler, Jaroslav Nešetřil (1999)

Commentationes Mathematicae Universitatis Carolinae

In this paper we investigate a class of problems permitting a good characterisation from the point of view of morphisms of oriented matroids. We prove several morphism-duality theorems for oriented matroids. These generalize LP-duality (in form of Farkas' Lemma) and Minty's Painting Lemma. Moreover, we characterize all morphism duality theorems, thus proving the essential unicity of Farkas' Lemma. This research helped to isolate perhaps the most natural definition of strong maps for oriented matroids....

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