We give the generating function for the index of integer lattice points, relative to a finite order ideal. The index is an important concept in the theory of border bases, an alternative to Gröbner bases.

We define the Laplacian operator on finite multicomplexes and give a formula for its spectra in the case of shifted multicomplexes.

We study the number of $k \times r$ plane partitions, weighted on the sum of the first row. Using Erhart reciprocity, we prove an identity for the generating function. For the special case $k=1$ this result follows from the classical theory of partitions, and for $k=2$ it was proved in Andersson-Bhowmik with another method. We give an explicit formula in terms of Young tableaux, and study the corresponding zeta-function. We give an application on the average orders of towers of abelian groups. In particular we prove that the number of isomorphism classes of ``subgroups of subgroups of ... ($k-1$ times) ... of abelian groups'' of order at most $N$ is asymptotic to $c_k N (\log N)^{k-1}$. This generalises results from Erd{\H o}s-Szekeres and Andersson-Bhowmik where the corresponding result was proved for $k=1$ and $k=2$.

Holm [Holm 04, Holm 02] studied modules of higher-order differential operators (generalizing derivations) on generic (central) hyperplane arrangements. We use his results to determine the Hubert series of these modules. We also give a conjecture about the Poincaré-Betti series, these are known for the module of derivations through the work of Yuzvinsky [Yuzvinsky 91] and Rose and Terao [Rose and Terao 91 ] © A K Peters, Ltd.

We show that certain two-dimensional, integrally closed monomial modules can be uniquely written as a countable product of isomorphic copies of simple integrally closed monomial ideals.

We study three different poset structures on the set of all compositions. In the first case, the covering relation consists of inserting a part of size one to the left or to the right, or increasing the size of some part by one. The resulting poset was studied by the author in "A poset classifying non-commutative term orders", and then in "Standard paths in another composition poset" where some results about generating functions for standard paths in this poset was established. The latter article was inspired by the work of Bergeron, Bousquet-M{\'e}lou and Dulucq on "Standard paths in the composition poset", where they studied a poset where there are additional cover relations which allows the insertion of a part of size one anywhere in the composition. Finally, following a suggestion by Richard Stanley we study yet a third which is an extension of the previous two posets. This poset is related to quasi-symmetric functions. For these posets, we study generating functions for saturated chains of fixed width k. We also construct ``labeled'' non-commutative generating functions and their associated languages.

We study how homogeneous ideals in the exterior algebra ? V over a finite-dimensional vector space V are minimally generated. In particular, we solve the following problems: Starting with an element pv of degree v, what is the maximum length l of a sequence pv, . . . , pv+l-l, with degpi = i, and such that pi is not in the ideal generated by pl, . . . , pi-l? What is the maximal possible number of minimal generators of degree d of a homogeneous ideal which does not contain all elements of degree d + 1? Our main tool is the Kruskal-Katona theorem.