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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Whitney number inequalities for geometric lattices


Authors: Thomas A. Dowling and Richard M. Wilson
Journal: Proc. Amer. Math. Soc. 47 (1975), 504-512
MathSciNet review: 0354422
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Abstract: Let $ L$ be a finite geometric lattice of rank $ r$, and for $ i = 0,1, \cdots ,r$, let $ {W_i}$ denote the number of elements of $ L$ with rank $ i$. For $ 1 \leq k \leq r - 2$, we have $ {W_1} + {W_2} + \cdots + {W_k} \leq {W_{r - k}} + \cdots + {W_{r - 2}} + {W_{r - 1}}$ with equality if and only if the lattice $ L$ is modular. We give two further results concerning matchings of lattice elements of rank $ \leq k$ into those of rank $ \geq r - k$, and observe that a middle term can be interpolated in the above inequality.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0354422-3
Article copyright: © Copyright 1975 American Mathematical Society