7 research outputs found
Face numbers of barycentric subdivisions of cubical complexes
The -polynomial of the barycentric subdivision of any -dimensional
cubical complex with nonnegative cubical -vector is shown to have only real
roots and to be interlaced by the Eulerian polynomial of type . This
result applies to barycentric subdivisions of shellable cubical complexes and,
in particular, to barycentric subdivisions of cubical convex polytopes and
answers affirmatively a question of Brenti, Mohammadi and Welker.Comment: Final version; minor changes, 11 page
Enumerative -theorems for the Veronese construction for formal power series and graded algebras
Let be a sequence of integers such that its generating
series satisfies for some
polynomial . For any we study the coefficient sequence of the
numerator polynomial
of the \textsuperscript{th} Veronese series . Under mild hypothesis we show that the vector of successive
differences of this sequence up to the \textsuperscript{th} entry is the -vector of a simplicial complex
for large . In particular, the sequence satisfies the consequences of the
unimodality part of the -conjecture. We give applications of the main result
to Hilbert series of Veronese algebras of standard graded algebras and the
-vectors of edgewise subdivisions of simplicial complexes
Face enumeration on simplicial complexes
Let be a closed triangulable manifold, and let be a
triangulation of . What is the smallest number of vertices that can
have? How big or small can the number of edges of be as a function of
the number of vertices? More generally, what are the possible face numbers
(-numbers, for short) that can have? In other words, what
restrictions does the topology of place on the possible -numbers of
triangulations of ?
To make things even more interesting, we can add some combinatorial
conditions on the triangulations we are considering (e.g., flagness,
balancedness, etc.) and ask what additional restrictions these combinatorial
conditions impose. While only a few theorems in this area of combinatorics were
known a couple of decades ago, in the last ten years or so, the field simply
exploded with new results and ideas. Thus we feel that a survey paper is long
overdue. As new theorems are being proved while we are typing this chapter, and
as we have only a limited number of pages, we apologize in advance to our
friends and colleagues, some of whose results will not get mentioned here.Comment: Chapter for upcoming IMA volume Recent Trends in Combinatoric