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

Abstract

Abstract.The doubly indexed Whitney numbers of a finite, ranked partially ordered set L are (the first kind) w;; = 2{/i(x', xj): x', xJ G L with ranks i, j] and (the second kind) W:j = the number of (x1, x') with x' < xJ. When L has a 0 element, the ordinary (simply indexed) Whitney numbers are Wj = w0j and W¡ = WQj = W:j. 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 ^-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.

The Tutte polynomial of a morphism of matroids I. Set-pointed matroids and matroid perspectives

“…This result generalizes in a self-dual way previous results of the literature on acyclic orientations of graphs [27], orientations of regular matroids [7], regions of real hyperplane arrangements [31], non Radon partitions of real spaces [5], acyclic reorientations of oriented matroids (or, equivalently, regions of pseudohyperplane arrangements), [17] [21], and bounded regions of hyperplane arrangements [15]. It has applications to the counting of containments of a given flat in facets of acyclic reorientations of an oriented matroid (i.e.…”

Baxter permutations and plane bipolar orientations

“…He proved that the coefficient of x 1 y 0 in T M (x, y), summed over all rooted non-separable planar maps M having n + 1 edges, m + 2 vertices, root-face of degree i + 1 and a root-vertex of degree j + 1, was given by (1). He was unaware that these numbers had been met before (and bear "his" name), and that the coefficient of x 1 y 0 in T M (x, y) is the number of bipolar orientations of M [24,20]. This number is also, up to a sign, the derivative of the chromatic polynomial of M, evaluated at 1 [26].…”

Fully Optimal Bases and the Active Bijection in Graphs, Hyperplane Arrangements, and Oriented Matroids

“…M. Las Vergnas has shown in [15] that t(M ; x, y) = i,j o i,j 2 −i−j x i y j where o i,j is the number of subsets A of E such that the oriented matroid − A M , obtained by reorientation of M on A, has dual-orientation activity i and orientation activity j. This second formula contains several results of the literature on counting acyclic orientations in graphs, regions in arrangements of hyperplanes and pseudohyperplanes, acyclic reorientations of oriented matroids with t(M ; 2, 0) [10] Comparing these two expressions for t(M ; x, y), we get the orientationbasis activity formula o i,j = 2 i+j b i,j for all i, j. The question arises of a bijective interpretation [15].…”