Margaret Readdy

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.

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.

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.

A Probabilistic Approach to the Descent Statistic

We present a probabilistic approach to studying the descent statistic based upon a two-variable probability density. This density is log concave and, in fact, satisfies a higher order concavity condition. From these properties we derive quadratic inequalities for the descent statistic. Using Fourier series, we give exact expressions for the Euler numbers and the alternating r-signed permutations. We also obtain a probabilistic interpretation of the sin function.

The Tchebyshev Transforms of the First and Second Kind

An in-depth study of the Tchebyshev transforms of the first and second kind of a poset is taken. The Tchebyshev transform of the first kind is shown to preserve desirable combinatorial properties, including EL-shellability and nonnegativity of the cd-index. When restricted to Eulerian posets, it corresponds to the Billera, Ehrenborg, and Readdy omega map of oriented matroids. The Tchebyshev transform of the second kind U is a Hopf algebra endomorphism on the space of quasisymmetric functions which, when restricted to Eulerian posets, coincides with Stembridge’s peak enumerator. The complete spectrum of U is determined, generalizing the work of Billera, Hsiao, and van Willigenburg. The type B quasisymmetric function of a poset is introduced and, like Ehrenborg’s classical quasisymmetric function of a poset, it is a comodule morphism with respect to the quasisymmetric functions QSym. Finally, similarities among the omega map, Ehrenborg’s r-signed Birkhoff transform, and the Tchebyshev transforms motivate a general study of chain maps which occur naturally in the setting of combinatorial Hopf algebras.

Mixed Volumes and Slices of the Cube

We give a combinatorial interpretation for the mixed volumes of two adjacent slices from the unit cube in terms of a refinement of the Eulerian numbers.

Homology of Newtonian Coalgebras

Given a Newtonian coalgebra w ea ssociate to it a chain complex. The homology groups of this Newtonian chain complex are computed for two important Newtonian coalgebras arising in the study of flag vectors of polytopes: Ra, band Rc, d� .T hehomology of Ra, bcorresponds to the homology of the boundary of the n-crosspolytope. In contrast, the homology of Rc, d� depends on the characteristic of the underlying ring R.I n thecase the ring has characteristic 2, the homology is computed via cubical complexes arising from distributive lattices. This paper ends with ac haracterization of the integer homology of Zc, d� .