 
Summary: CUBICAL SUBDIVISIONS AND LOCAL hVECTORS
CHRISTOS A. ATHANASIADIS
Abstract. Face numbers of triangulations of simplicial complexes were studied by Stan
ley by use of his concept of a local hvector. It is shown that a parallel theory exists for
cubical subdivisions of cubical complexes, in which the role of the hvector of a simplicial
complex is played by the (short or long) cubical hvector of a cubical complex, defined by
Adin, and the role of the local hvector of a triangulation of a simplex is played by the
(short or long) cubical local hvector of a cubical subdivision of a cube. The cubical local
hvectors 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
hvector (equivalently, of the hpolynomial) of , changes under simplicial subdivisions of
various types; see [19, Chapter II] for the importance of hvectors in the combinatorics of
simplicial complexes.
A key result in this study is a formula [17, Equation (2)] which expresses the hpolynomial
of a simplicial subdivision of a pure simplicial complex as the sum of the hpolynomial
