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CUBICAL SUBDIVISIONS AND LOCAL h-VECTORS CHRISTOS A. ATHANASIADIS
 

Summary: CUBICAL SUBDIVISIONS AND LOCAL h-VECTORS
CHRISTOS A. ATHANASIADIS
Abstract. Face numbers of triangulations of simplicial complexes were studied by Stan-
ley by use of his concept of a local h-vector. It is shown that a parallel theory exists for
cubical subdivisions of cubical complexes, in which the role of the h-vector of a simplicial
complex is played by the (short or long) cubical h-vector of a cubical complex, defined by
Adin, and the role of the local h-vector of a triangulation of a simplex is played by the
(short or long) cubical local h-vector of a cubical subdivision of a cube. The cubical local
h-vectors are defined in this paper and are shown to share many of the properties of their
simplicial counterparts. Generalizations to subdivisions of locally Eulerian posets are also
discussed.
1. Introduction
Simplicial subdivisions (or triangulations) of simplicial complexes were studied from an
enumerative point of view by Stanley [17]. Specifically, the paper [17] is concerned with the
way in which the face enumeration of a simplicial complex , presented in the form of the
h-vector (equivalently, of the h-polynomial) of , changes under simplicial subdivisions of
various types; see [19, Chapter II] for the importance of h-vectors in the combinatorics of
simplicial complexes.
A key result in this study is a formula [17, Equation (2)] which expresses the h-polynomial
of a simplicial subdivision of a pure simplicial complex as the sum of the h-polynomial

  

Source: Athanasiadis, Christos - Department of Mathematics, University of Athens

 

Collections: Mathematics