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

Greene, Curtis; Zaslavsky, Thomas

doi:10.1090/S0002-9947-1983-0712251-1

On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs
HTML articles powered by AMS MathViewer

by Curtis Greene and Thomas Zaslavsky PDF

Trans. Amer. Math. Soc. 280 (1983), 97-126 Request permission

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.

References

G. L. Alexanderson and John E. Wetzel, Dissections of a plane oval, Amer. Math. Monthly 84 (1977), no. 6, 442–449. MR 513837, DOI 10.2307/2321901
G. L. Alexanderson and John E. Wetzel, Arrangements of planes in space, Discrete Math. 34 (1981), no. 3, 219–240. MR 613402, DOI 10.1016/0012-365X(81)90002-9
Gerald Berman, The dichromate and orientations of a graph, Canadian J. Math. 29 (1977), no. 5, 947–956. MR 469810, DOI 10.4153/CJM-1977-095-1
Robert G. Bland and Michel Las Vergnas, Orientability of matroids, J. Combinatorial Theory Ser. B 24 (1978), no. 1, 94–123. MR 485461, DOI 10.1016/0095-8956(78)90080-1
Thomas H. Brylawski, A decomposition for combinatorial geometries, Trans. Amer. Math. Soc. 171 (1972), 235–282. MR 309764, DOI 10.1090/S0002-9947-1972-0309764-6
T. H. Brylawski, A combinatorial perspective on the Radon convexity theorem, Geometriae Dedicata 5 (1976), no. 4, 459–466. MR 440465, DOI 10.1007/BF00150777
Beniamino Segre, Teorie combinatorie, Colloquio Internazionale sulle Teorie Combinatorie (Roma, 1973) Atti dei Convegni Lincei, No. 17, Accad. Naz. Lincei, Rome, 1976, pp. 7–12. MR 0434827
Louis Comtet, Advanced combinatorics, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974. The art of finite and infinite expansions. MR 0460128
H. S. M. Coxeter, The classification of zonohedra by means of projective diagrams, J. Math. Pures Appl. (9) 41 (1962), 137–156. MR 141004
Henry H. Crapo, A higher invariant for matroids, J. Combinatorial Theory 2 (1967), 406–417. MR 215744

Radon’s theorem revisited

81f

Jon Folkman and Jim Lawrence, Oriented matroids, J. Combin. Theory Ser. B 25 (1978), no. 2, 199–236. MR 511992, DOI 10.1016/0095-8956(78)90039-4

Acyclic orientations

Jeanne W. Kerr and John E. Wetzel, Dissections of a triangular prism, Geometriae Dedicata 4 (1975), no. 2, /3/4, 279–289. MR 398863, DOI 10.1007/BF00148762
Michel Las Vergnas, Matroïdes orientables, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), Ai, A61–A64 (French, with English summary). MR 371692
Michel Las Vergnas, Acyclic and totally cyclic orientations of combinatorial geometries, Discrete Math. 20 (1977/78), no. 1, 51–61. MR 462993, DOI 10.1016/0012-365X(77)90042-5
Michel Las Vergnas, Sur les activités des orientations d’une géométrie combinatoire, Cahiers Centre Études Rech. Opér. 20 (1978), no. 3-4, 293–300 (French). MR 543171
Michel Las Vergnas, Convexity in oriented matroids, J. Combin. Theory Ser. B 29 (1980), no. 2, 231–243. MR 586435, DOI 10.1016/0095-8956(80)90082-9
P. McMullen, On zonotopes, Trans. Amer. Math. Soc. 159 (1971), 91–109. MR 279689, DOI 10.1090/S0002-9947-1971-0279689-2
P. McMullen, Transforms, diagrams and representations, Contributions to geometry (Proc. Geom. Sympos., Siegen, 1978) Birkhäuser, Basel-Boston, Mass., 1979, pp. 92–130. MR 568496

Théorie des éléments saillants d’une suite d’observations

44

Gian-Carlo Rota, On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340–368 (1964). MR 174487, DOI 10.1007/BF00531932
Richard P. Stanley, Acyclic orientations of graphs, Discrete Math. 5 (1973), 171–178. MR 317988, DOI 10.1016/0012-365X(73)90108-8
Thomas Zaslavsky, Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Mem. Amer. Math. Soc. 1 (1975), no. issue 1, 154, vii+102. MR 357135, DOI 10.1090/memo/0154
Thomas Zaslavsky, Counting the faces of cut-up spaces, Bull. Amer. Math. Soc. 81 (1975), no. 5, 916–918. MR 400066, DOI 10.1090/S0002-9904-1975-13885-7
Thomas Zaslavsky, A combinatorial analysis of topological dissections, Advances in Math. 25 (1977), no. 3, 267–285. MR 446994, DOI 10.1016/0001-8708(77)90076-7
Thomas Zaslavsky, Arrangements of hyperplanes; matroids and graphs, Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979) Congress. Numer., XXIII–XXIV, Utilitas Math., Winnipeg, Man., 1979, pp. 895–911. MR 561105

Signed graphs

4

Erratum

5

Orientation of signed graphs

Thomas Zaslavsky, Signed graph coloring, Discrete Math. 39 (1982), no. 2, 215–228. MR 675866, DOI 10.1016/0012-365X(82)90144-3