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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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The slimmest geometric lattices
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by Thomas A. Dowling and Richard M. Wilson
Trans. Amer. Math. Soc. 196 (1974), 203-215
DOI: https://doi.org/10.1090/S0002-9947-1974-0345849-8

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.
References
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Bibliographic Information
  • © Copyright 1974 American Mathematical Society
  • 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