Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

The slimmest geometric lattices


Authors: Thomas A. Dowling and Richard M. Wilson
Journal: Trans. Amer. Math. Soc. 196 (1974), 203-215
MSC: Primary 05B35
DOI: https://doi.org/10.1090/S0002-9947-1974-0345849-8
MathSciNet review: 0345849
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Whitney numbers of a finite geometric lattice L of rank r are the numbers $ {W_k}$ of elements of rank k and the coefficients $ {w_k}$ of the characteristic polynomial of L, for $ 0 \leq k \leq r$. We establish the following lower bounds for the $ {W_k}$ and the absolute values $ w_k^ + = {( - 1)^k}{w_k}$ and describe the lattices for which equality holds (nontrivially) in each case:

$\displaystyle {W_k} \geq \left( {\begin{array}{*{20}{c}} r \hfill & - \hfill & ... ... \left( {\begin{array}{*{20}{c}} r \hfill \\ k \hfill \\ \end{array} } \right),$

where $ n = {W_1}$ is the number of points of L.

References [Enhancements On Off] (What's this?)

  • [1] J. G. Basterfield and L. M. Kelly, A characterization of sets of n points which determine n hyperplanes, Proc. Cambridge Philos. Soc. 64 (1968), 585-588. MR 38 #2040. MR 0233719 (38:2040)
  • [2] G. Birkhoff, Lattice theory, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, R. I., 1967. MR 37 #2638. MR 0227053 (37:2638)
  • [3] J. E. Blackburn, H. H. Crapo and D. A. Higgs, A catalogue of combinatorial geometries, University of Waterloo, Waterloo, Ontario, 1969.
  • [4] H. H. Crapo and G.-C. Rota, On the foundations of combinatorial theory: Combinatorial geometries, Preliminary edition, M. I. T. Press, Cambridge, Mass., 1970. MR 45 #74. MR 0290980 (45:74)
  • [5] T. A. Dowling and R. M. Wilson, Whitney number inequalities for geometric lattices, Proc. Amer. Math. Soc. (to appear). MR 0354422 (50:6900)
  • [6] C. Greene, A rank inequality for finite geometric lattices, J. Combinatorial Theory 9 (1970), 357-364 MR 42 # 1727. MR 0266824 (42:1727)
  • [7] -, An inequality for the Möbius function of a geometric lattice, Proc. Conference on Möbius Algebras (H. Crapo and G. Roulet, ed.), University of Waterloo, 1971. MR 0349444 (50:1938)
  • [8] L. H. Harper, Stirling behavior is asymptotically normal, Ann. Math. Statist. 38 (1967), 410-414 MR 35 #2312. MR 0211432 (35:2312)
  • [9] E. H. Lieb, Concavity properties and a generating function for Stirling numbers, J. Combinatorial Theory 5 (1968), 203-206. MR 37 #6195. MR 0230635 (37:6195)
  • [10] G.-C. Rota, On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340-368. MR 30 #4688. MR 0174487 (30:4688)
  • [11] P. Young, U. S. R. Murty and J. Edmonds, Equicardinal matroids and matroid-designs, Proc. Second Chapel Hill Conf. on Combinatorial Mathematics and its Applications, Univ. of North Carolina, Chapel Hill, N. C., 1970, pp. 498-542. MR 42 #1685. MR 0266782 (42:1685)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 05B35

Retrieve articles in all journals with MSC: 05B35


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0345849-8
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society