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A geometric LittlewoodRichardson rule
 ANNALS OF MATHEMATICS, 164 (2006), 371422
, 2006
"... We describe a geometric LittlewoodRichardson rule, interpreted as deforming the intersection of two Schubert varieties into the union of Schubert varieties. There are no restrictions on the base eld, and all multiplicities arising are 1; this is important for applications. This rule should be see ..."
Abstract

Cited by 54 (4 self)
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We describe a geometric LittlewoodRichardson rule, interpreted as deforming the intersection of two Schubert varieties into the union of Schubert varieties. There are no restrictions on the base eld, and all multiplicities arising are 1; this is important for applications. This rule should be seen as a generalization of Pieri's rule to arbitrary Schubert classes, by way of explicit homotopies. It has straightforward bijections to other LittlewoodRichardson rules, such as tableaux, and Knutson and Tao's puzzles. This gives the rst geometric proof and interpretation of the LittlewoodRichardson rule. Geometric consequences are described here and in [V2], [KV1], [KV2], [V3]. For example, the rule also has an interpretation in Ktheory, suggested by Buch, which gives an extension of puzzles to Ktheory.
ON LINEAR TRANSFORMATIONS PRESERVING THE PÓLYA FREQUENCY PROPERTY
"... We prove that certain linear operators preserve the Pólya frequency property and realrootedness, and apply our results to settle some conjectures and open problems in combinatorics proposed by Bóna, Brenti and ReinerWelker. ..."
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Cited by 29 (5 self)
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We prove that certain linear operators preserve the Pólya frequency property and realrootedness, and apply our results to settle some conjectures and open problems in combinatorics proposed by Bóna, Brenti and ReinerWelker.
The independence polynomial of a graph  a survey
, 2005
"... A stable (or independent) set in a graph is a set of pairwise nonadjacent vertices. The stability number α(G) is the size of a maximum stable set in the graph G. There are three different kinds of structures that one can see observing behavior of stable sets of a graph: the enumerative structure, t ..."
Abstract

Cited by 17 (5 self)
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A stable (or independent) set in a graph is a set of pairwise nonadjacent vertices. The stability number α(G) is the size of a maximum stable set in the graph G. There are three different kinds of structures that one can see observing behavior of stable sets of a graph: the enumerative structure, the intersection structure, and the exchange structure. The independence polynomial of G I(G; x) = α(G) � k=0 skx k = s0 + s1x + s2x 2 +... + sα(G)x α(G), defined by Gutman and Harary (1983), is a good representative of the enumerative structure (sk is the number of stable sets of cardinality k in a graph G). One of the most general approaches to graph polynomials was proposed by Farrell (1979) in his theory of Fpolynomials of a graph. According to Farrell, any such polynomial corresponds to a strictly prescribed family of connected subgraphs of the respective graph. For the matching polynomial of a graph G, this family consists of all the edges of G, for the independence polynomial of G, this family includes all the stable sets of G. In fact, various aspects of combinatorial information concerning a graph is stored in the coefficients of a specific graph polynomial. In this paper, we survey the most important results referring the independence polynomial of a graph.