We give a series of successively weaker conjectures on the cohomology ring of the Grassmannian, s... more We give a series of successively weaker conjectures on the cohomology ring of the Grassmannian, starting with the Hilbert series of a certain natural filtration.
In this paper we show that the set of closure relations on a nite poset P forms a supersolvable l... more In this paper we show that the set of closure relations on a nite poset P forms a supersolvable lattice, as suggested by Rota. Furthermore this lattice is dually isomorphic to the lattice of closed sets in a convex geometry (in the sense of Edelman and Jamison EJ]). We also characterize the modular elements of this lattice (when P has a greatest element) and compute its characteristic polynomial.
Non-Crossing Partitions For Classical Reflection Groups
... 4} {-4}. To reverse the bijection, given n E NCB(n), find a non-zero block which ... Sim-ilar... more ... 4} {-4}. To reverse the bijection, given n E NCB(n), find a non-zero block which ... Sim-ilarly, given a signed permutation w in the hyperoctahedral group Bn acting as per-mutations ... The 212-avoiding elements of B~, are in natural bijection with content-labelled shifted skew shapes ...
3 A ring is said to be discrete provided that one can constructively decide whether an element of... more 3 A ring is said to be discrete provided that one can constructively decide whether an element of the ring equals zero.
For finite reflection groups of types A and B, we determine the diameter of the graph whose verti... more For finite reflection groups of types A and B, we determine the diameter of the graph whose vertices are reduced words for the longest element and whose edges are braid relations. This is deduced from a more general theorem that applies to supersolvable hyperplane arrangements.
The Electronic Journal of Combinatorics, Nov 2, 2009
We present two new problems on lower bounds for Betti numbers of the minimal free resolution for ... more We present two new problems on lower bounds for Betti numbers of the minimal free resolution for monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are quadratic and bihomogeneous with respect to two variable sets, but gives a more finely graded lower bound.
We survey the generalized Baues problem of Billera and Sturm- fels. The problem is one of discret... more We survey the generalized Baues problem of Billera and Sturm- fels. The problem is one of discrete geometry and topology, and asks about the topology of the set of subdivisions of a certain kind of a convex poly- tope. Along with a discussion of most of the known results, we survey the motivation for the problem and its relation to
3 A ring is said to be discrete provided that one can constructively decide whether an element of... more 3 A ring is said to be discrete provided that one can constructively decide whether an element of the ring equals zero.
We present two new problems on lower bounds for resolution Betti numbers of monomial ideals gener... more We present two new problems on lower bounds for resolution Betti numbers of monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are quadratic and bihomogeneous with respect to two variable sets, but gives a more finely graded lower bound. These problems are solved for certain classes of ideals that generalize (in two different directions) the edge ideals of threshold graphs and Ferrers graphs. In the process, we produce particularly simple cellular linear resolutions for strongly stable and squarefree strongly stable ideals generated in a fixed degree, and combinatorial interpretations for the Betti numbers of other classes of ideals, all of which are independent of the coefficient field.
A Catalan triangulation of the Mobius band is an abstract simplicial complex triangulating the Mo... more A Catalan triangulation of the Mobius band is an abstract simplicial complex triangulating the Mobius band which uses no interior vertices, and has verticeslabelled 1, 2, .. . , n in order as one traverses the boundary. We prove two results about the structure of this set, analogous to well-known results for Catalan triangulations of the disk. The first is a generating function for Catalan triangulations of M having n vertices, and the second is that any two such triangulations are connected by a sequence of diagonal-flips.
Uploads
Papers by Victor Reiner