Linear inequalities for flags in graded posets

The closure of the convex cone generated by all flag $f$-vectors of graded posets is shown to be polyhedral. In particular, we give the facet inequalities to the polar cone of all nonnegative chain-enumeration functionals on this class of posets. These are in one-to-one correspondence with antichains of intervals on the set of ranks and thus are counted by Catalan numbers. Furthermore, we prove that the convolution operation introduced by Kalai assigns extreme rays to pairs of extreme rays in most cases. We describe the strongest possible inequalities for graded posets of rank at most 5

Constructions and complexity of secondary polytopes

AbstractThe secondary polytope Σ(A) of a configuration A of n points in affine (d − 1)-space is an (n − d)-polytope whose vertices correspond to regular triangulations of conv(A). In this article we present three constructions of Σ(A) and apply them to study various geometric, combinatorial, and computational properties of secondary polytopes. The first construction is due to Gel'fand, Kapranov, and Zelevinsky, who used it to describe the face lattice of Σ(A). We introduce the universal polytope u(A) ⊂ ΛdRn, a combinatorial object depending only on the oriented matroid of A. The secondary Σ(A) can be obtained as the image of u(A) under a canonical linear map onto Rn. The third construction is based upon Gale transforms or oriented matroid duality. It is used to analyze the complexity of computing Σ(A) and to give bounds in terms of n and d for the number of faces of Σ(A)

A quasisymmetric function for matroids

AbstractA new isomorphism invariant of matroids is introduced, in the form of a quasisymmetric function. This invariant: •defines a Hopf morphism from the Hopf algebra of matroids to the quasisymmetric functions, which is surjective if one uses rational coefficients;•is a multivariate generating function for integer weight vectors that give minimum total weight to a unique base of the matroid;•is equivalent, via the Hopf antipode, to a generating function for integer weight vectors which keeps track of how many bases minimize the total weight;•behaves simply under matroid duality;•has a simple expansion in terms of P-partition enumerators;•is a valuation on decompositions of matroid base polytopes. This last property leads to an interesting application: it can sometimes be used to prove that a matroid base polytope has no decompositions into smaller matroid base polytopes. Existence of such decompositions is a subtle issue arising from the work of Lafforgue, where lack of such a decomposition implies that the matroid has only a finite number of realizations up to scalings of vectors and overall change-of-basis