Richard Ehrenborg

Apolarity and Canonical Forms for Homogeneous Polynomials

Kare Bernt Lindstrom,Gratulerar pa Din sextioars dag. Vi hoppas att denna kommer falla Dig i smaken.Vi borjar med en kort studie i algebraiska matroider, och fortsatter med att bevisa relationen mellan Jacobianen av en mangd algebraiska funktioner och deras algebraiska oberoende. Med detta resulat bevisar vi de tva huvudsatserna, som behandlar kanoniska former. Dessa satser reducerar fragan om en form ar kanonisk for homogena polynom i q variabler och av grad p till att undersoka om ett homogent linjart ekvations system har bara den triviala losningen. Genom att anvanda apolaritet kan detta linjara system enkelt beskrivas. Till sist ger vi en mangfald av exempel av kanoniska former for homogena polynom.

Coproducts and the cd-Index

The linear span of isomorphism classes of posets, P, has a Newtonian coalgebra structure. We observe that the ab-index is a Newtonian coalgebra map from the vector space P to the algebra of polynomials in the noncommutative variables a and b. This enables us to obtain explicit formulas showing how the cd-index of the face lattice of a convex polytope changes when taking the pyramid and the prism of the polytope and the corresponding operations on posets. As a corollary, we have new recursion formulas for the cd-index of the Boolean algebra and the cubical lattice. Moreover, these operations also have interpretations for certain classes of permutations, including simsun and signed simsun permutations. We prove an identity for the shelling components of the simplex. Lastly, we show how to compute the ab-index of the Cartesian product of two posets given the ab-indexes of each poset.

The Hankel Determinant of Exponential Polynomials

The purpose of this note is two-fold. First we present evaluations of Hankel determinants of sequences of combinatorial interest related to partitions and permutations. Many such computations have been carried out by Radoux in his sequence of papers [2]–[5]. His proof methods include using a functional identity due to Sylvester and factoring the Hankel matrix. Second, unlike Radoux, we instead give bijective proofs to reveal the underlying structure of these identities.

Thec-2d-Index of Oriented Matroids

We obtain an explicit method to compute thecd-index of the lattice of regions of an oriented matroid from theab-index of the corresponding lattice of flats. Since thecd-index of the lattice of regions is a polynomial in the ring Z(c,2d), we call it thec-2d-index. As an application we obtain a zonotopal analogue of a conjecture of Stanley: among all zonotopes the cubical lattice has the smallestc-2d-index coefficient-wise. We give a new combinatorial description for thec-2d-index of the cubical lattice and thecd-index of the Boolean algebra in terms of all the permutations in the symmetric groupSn. Finally, we show that only two-thirds of the?(S)sof the lattice of flats are needed for thec-2d-index computation.

The topology of the independence complex

We introduce a large self-dual class of simplicial complexes for which we show that each member complex is contractible or homotopy equivalent to a sphere. Examples of complexes in this class include independence and dominance complexes of forests, pointed simplicial complexes, and their combinatorial Alexander duals.

Juggling and applications to q -analogues

We consider juggling patterns where the juggler can only catch and throw one ball at a time, and patterns where the juggler can handle many balls at the same time. Using a crossing statistic, we obtain explicit q-enumeration formulas. Our techniques give a natural combinatorial interpretation of the q-Stirling numbers of the second kind and a bijective proof of an identity of Carlitz. By generalizing these techniques, we give a bijective proof of a q-identity involving unitary compositions due to Haglund. Also, juggling patterns enable us to easily compute the Poincare series of the affine Weyl group Ad−1.

Affine and Toric Hyperplane Arrangements

We extend the Billera–Ehrenborg–Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky’s fundamental results on the number of regions.

The r-cubical lattice and a generalization of the cd-index

Abstract In this paper we generalize thecd-index of the cubical lattice to anr-cd-index, which we denote byΨ(r). The coefficients ofΨ(r) enumerate augmented Andrer-signed permutations, a generalization of Purtills work relating thecd-index of the cubical lattice and signed Andre permutations. As an application we use ther-cd-index to determine that the extremal configuration which maximizes the Mobius function of arbitrary rank selections, where all theris are greater than one, is the odd alternating ranks, {1, 3, 5, ...}.

The cd-Index of Zonotopes and Arrangements

We investigate a special class of polytopes, the zonotopes, and show that their flag f-vectors satisfy only the affine relations fulfilled by flag f-vectors of all polytopes. In addition, we determine the lattice spanned by flag f-vectors of zonotopes. By duality, these results apply as well to the flag f-vectors of central arrangements of hyperplanes.

The toric ℎ-vectors of partially ordered sets

First published in TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 352(10), published by the American Mathematical Society.