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Face Numbers of Polytopes, Posets, and Complexes

出版	University of Washington Libraries, 2022
URL	http://books.google.com.hk/books?id=SohT0AEACAAJ&hl=&source=gbs_api

註釋A key tool that combinatorialists use to study simplicial complexes and polytopes is the f-vector (or face vector), which records the number of faces of each dimension. In order to better understand the face numbers, relations involving both equalities and inequalities on f-vectors have been extensively studied. In this dissertation we discuss the author's contributions to these topics. The classical Dehn--Sommerville relations assert that the h-vectorof an Eulerian simplicial complex is symmetric. In Chapter 2, we establish three generalizations of the Dehn--Sommerville relations: one for the h-vectors of pure simplicial complexes, another one for the flag h-vectors of balanced simplicial complexes and graded posets, and yet another one for the toric h-vectors of graded posets with restricted singularities. In all of these cases, we express any failure of symmetry in terms of "errors coming from the links." For simplicial complexes, this further extends Klee's semi-Eulerian relations. In Chapters 3 and 4, we change our focus from equalities to inequalities on f-vectors. In 1967, Grünbaum conjectured that any d-dimensional polytope with d+s