96 research outputs found
Arithmetic of marked order polytopes, monotone triangle reciprocity, and partial colorings
For a poset P, a subposet A, and an order preserving map F from A into the
real numbers, the marked order polytope parametrizes the order preserving
extensions of F to P. We show that the function counting integral-valued
extensions is a piecewise polynomial in F and we prove a reciprocity statement
in terms of order-reversing maps. We apply our results to give a geometric
proof of a combinatorial reciprocity for monotone triangles due to Fischer and
Riegler (2011) and we consider the enumerative problem of counting extensions
of partial graph colorings of Herzberg and Murty (2007).Comment: 17 pages, 10 figures; V2: minor changes (including title); V3:
examples included (suggested by referee), to appear in "SIAM Journal on
Discrete Mathematics
On f- and h- vectors of relative simplicial complexes
A relative simplicial complex is a collection of sets of the form , where are simplicial complexes.
Relative complexes played key roles in recent advances in algebraic, geometric,
and topological combinatorics but, in contrast to simplicial complexes, little
is known about their general combinatorial structure. In this paper, we address
a basic question in this direction and give a characterization of -vectors
of relative (multi)complexes on a ground set of fixed size. On the algebraic
side, this yields a characterization of Hilbert functions of quotients of
homogeneous ideals over polynomial rings with a fixed number of indeterminates.
Moreover, we characterize -vectors of fully Cohen--Macaulay relative
complexes as well as -vectors of Cohen--Macaulay relative complexes with
minimal faces of given dimensions. The latter resolves a question of Bj\"orner.Comment: accepted for publication in Algebraic Combinatoric
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