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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
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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.

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