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.     

Compact Compatible Topologies for Posets and Graphs

A topology on the vertex set of a comparability graph G is said to be compatible (respectively, weakly compatible) with G if each induced subgraph (respectively, each finite induced subgraph) is topologically connected if and only it it is graph-connected; a weakly compatible topology on the vertex set of a graph completely determines the graph structure. We consider here the problem of deciding whether or not a comparability graph has a compact compatible or weakly compatible topology and in the case of graphs with small cycles, hence in the case of trees, we give a characterization.     

Polygon Posets and the Weak Order of Coxeter Groups

We explore the connection between polygon posets, which is a class of ranked posets with an edge-labeling which satisfies certain polygon properties, and the weak order of Coxeter groups. We show that every polygon poset is isomorphic to a join ideal in the weak order, and for Coxeter groups where no pair of generators have infinite order the converse is also true.The class of polygon posets is seen to include the class of generalized quotients defined by Björner and Wachs, while itself being included in the class of alternative generalized quotients also considered by these authors. By studying polygon posets we are then able to answer an open question about common properties of these two classes.     

5元素集合上T_0拓扑总数的计算

利用有限偏序集上的几个重要结果并借助于拓扑空间对应的特殊化序与拓扑之间的关系计算得出5元素集合上T0拓扑总数为4231,拓扑总数为6942.     

Deletion-restriction for subspace arrangements

Let A be a subspace arrangement in V with a designated maximal affine subspace A₀. Let A^′=A?{A₀} be the deletion of A₀ from A and A^″={A∩A₀|A∩A₀≠∅} be the restriction of A to A₀. Let M=V?_?A∈AA be the complement of A in V. If A is an arrangement of complex affine hyperplanes, then there is a split short exact sequence, 0→Hk(M^′)→Hk(M)→Hk+1−codimR(A₀)(M^″)→0. In this paper, we determine conditions for when the triple (A,A^′,A^″) of arrangements of affine subspaces yields the above split short exact sequence. We then generalize the no-broken-circuit basis nbc of Hk(M) for hyperplane arrangements to deletion-restriction subspace arrangements.     

Linear Discrepancy of the Product of Two Chains

The linear discrepancy of a partially ordered set P = (X, ≺) is the minimum integer l such that ∣f(a) − f(b)∣ ≤ l for any injective isotone and any pair of incomparable elements a, b in X. It measures the degree of difference of P from a chain. Despite of increasing demands to the applications, the discrepancies of just few simple partially ordered sets are known. In this paper, we obtain the linear discrepancy of the product of two chains. For this, we firstly give a lower bound of the linear discrepancy and then we construct injective isotones on the product of two chains, which show that the obtained lower bound is tight.     

Shelling the coset poset

It is shown that the coset lattice of a finite group has shellable order complex if and only if the group is complemented. Furthermore, the coset lattice is shown to have a Cohen-Macaulay order complex in exactly the same conditions. The group theoretical tools used are relatively elementary, and avoid the classification of finite simple groups and of minimal finite simple groups.     

A matroid-friendly basis for the quasisymmetric functions

A new Z-basis for the space of quasisymmetric functions (QSym, for short) is presented. It is shown to have nonnegative structure constants, and several interesting properties relative to the quasisymmetric functions associated to matroids by the Hopf algebra morphism F of Billera, Jia, and Reiner [L.J. Billera, N. Jia, V. Reiner, A quasisymmetric function for matroids, arXiv:math.CO/0606646]. In particular, for loopless matroids, this basis reflects the grading by matroid rank, as well as by the size of the ground set. It is shown that the morphism F distinguishes isomorphism classes of rank two matroids, and that decomposability of the quasisymmetric function of a rank two matroid mirrors the decomposability of its base polytope. An affirmative answer to the Hilbert basis question raised in [L.J. Billera, N. Jia, V. Reiner, A quasisymmetric function for matroids, arXiv:math.CO/0606646] is given.     

Polyspherical Coordinate Systems on Orbit Spaces with Applications to Biomolecular Shape

A general theory of molecular internal coordinates of valence type is presented based on the concept of a Z-system. The Z-system can be considered as a discrete mathematical generalization of the Z-matrix (a molecular geometry file format familiar to chemists) which avoids the principal disadvantage of Z-matrices. Z-matrices are usually only employed for small molecules because there is no easy way to glue two Z-matrices together to get the Z-matrix of a larger molecule. It is shown that Z-matrices are simply Z-systems together with additional extraneous structures and that the Z-systems for any two molecules can be naturally glued together to obtain a Z-system for the combined molecule. A general mathematical framework suitable for the detailed study of molecular geometry is introduced and applied to five and six-membered molecular rings. A classification of shapes of hexagons with opposite sides and angles congruent is given with explicit parameterizations of the flexible and rigid solutions. The entire mathematical formalism generalizes to a theory of polyspherical coordinate systems on orbit spaces of the group of -dimensional rigid motions acting on finite collections of points in -dimensional Euclidean space. The -dimensional Z-system is a new discrete structure related to abstract simplicial complexes, graded posets, and iterated line graphs. Complete proofs of all the -dimensional results are given, and connections to other areas of mathematics are noted.