The slimmest geometric lattices

Dowling, Thomas A.; Wilson, Richard M.

doi:10.1090/S0002-9947-1974-0345849-8

The slimmest geometric lattices
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by Thomas A. Dowling and Richard M. Wilson PDF

Trans. Amer. Math. Soc. 196 (1974), 203-215 Request permission

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: \[ {W_k} \geq \left ( {\begin {array}{*{20}{c}} r \hfill & - \hfill & 2 \hfill \\ k \hfill & - \hfill & 1 \hfill \\ \end {array} } \right )(n - r) + \left ( {\begin {array}{*{20}{c}} r \hfill \\ k \hfill \\ \end {array} } \right ),\quad w_k^ + \geq \left ( {\begin {array}{*{20}{c}} r \hfill & - \hfill & 1 \hfill \\ k \hfill & - \hfill & 1 \hfill \\ \end {array} } \right )(n - r) + \left ( {\begin {array}{*{20}{c}} r \hfill \\ k \hfill \\ \end {array} } \right ),\] where $n = {W_1}$ is the number of points of L.

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