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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Authors: Curtis Greene and Thomas Zaslavsky
Journal: Trans. Amer. Math. Soc. 280 (1983), 97-126
MSC: Primary 05B35; Secondary 05C20, 51M20, 52A25
MathSciNet review: 712251
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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.

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Keywords: Matroid, combinatorial geometry, geometric lattice, Whitney numbers, arrangement of hyperplanes, partition of space, zonotope, Radon partition, digraph, oriented graph, acyclic orientation, totally cyclic orientation, signed graph
Article copyright: © Copyright 1983 American Mathematical Society