Symmetric Simplicial Pseudoline Arrangements A simplicial arrangement of pseudolines is a collection of topological lines in the projective plane where each region that is formed is triangular. This paper refines and develops David Eppsteinu0027s notion of a kaleidoscope construction for symmetric pseudoline arrangements to construct and analyze several infinite families of simplicial pseudoline arrangements with high degrees of geometric symmetry. In particular, all simplicial pseudoline arrangements with the symmetries of a regular kgon and three symmetry classes of pseudolines, consisting of the mirrors of the kgon and two other symmetry classes, plus sometimes the line at infinity, are classified, and other interesting families (with more symmetry classes of pseudolines) are discussed.

Homotopy type of skeleta of the flag complex over a finite vector space and generalized Galois numbers Let (mathscr F) stand for the flag complex associated to the lattice of proper subspaces of a finitedimensional vector space V. This paper aims at giving a (discrete) Morse theoretical proof of the fact that the kth skeleton of (mathscr F) is homotopy equivalent to a wedge of spheres of dimension (min k,dim (mathscr F)). The tight control provided by Morse theoretic methods (through an explicit discrete gradient field) allows us to give a formula for the number of spheres appearing in each of these wedge summands. As an application, we derive an explicit formula for the number of flags on V of a given dimension, i.e., the number of simplices in (mathscr F) of the given dimension. Rather than depending on generalized Galois numbers, our formula for flags is given in terms of weighted inversion statistics of the symmetric group.