On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs

Abstract:

The doubly indexed Whitney numbers of a finite, ranked partially ordered set $L$ are (the first kind) ${w_{ij}} = \sum {\{ \mu ({x^i},{x^j}):{x^i},{x^j} \in L}$ with ranks $i,j\}$ and (the second kind) ${W_{ij}} =$ the number of $({x^i},{x^j})$ with ${x^i} \leqslant {x^j}$. When $L$ has a $0$ element, the ordinary (simply indexed) Whitney numbers are ${w_j} = {w_{0j}}$ and ${W_j} = {W_{0j}} = {W_{jj}}$ . Building on work of Stanley and Zaslavsky we show how to interpret the magnitudes of Whitney numbers of geometric lattices and semilattices arising in geometry and graph theory. For example: The number of regions, or of $k$-dimensional faces for any $k$, of an arrangement of hyperplanes in real projective or affine space, that do not meet an arbitrary hyperplane in general position. The number of vertices of a zonotope $P$ inside the visible boundary as seen from a distant point on a generating line of $P$. The number of non-Radon partitions of a Euclidean point set not due to a separating hyperplane through a fixed point. The number of acyclic orientations of a graph (Stanley’s theorem, with a new, geometrical proof); the number with a fixed unique source; the number whose set of increasing arcs (in a fixed ordering of the vertices) has exactly $q$ sources (generalizing Rényi’s enumeration of permutations with $q$ "outstanding" elements). The number of totally cyclic orientations of a plane graph in which there is no clockwise directed cycle. The number of acyclic orientations of a signed graph satisfying conditions analogous to an unsigned graph’s having a unique source.