Whitney Numbers of Combinatorial Geometries and Higher-Weight Dowling Lattices
Alberto Ravagnani · SIAM Journal on Applied Algebra and Geometry · 2022
We study the Whitney numbers of the first kind of combinatorial geometries, in connection with the theory of error-correcting codes. The first part of the paper is devoted to general results relating the Möbius functions of nested atomistic lattices, extending some classical theorems in combinatorics. We then specialize our results to restriction geometries, i.e., to sublattices $\mathcal{A}$ of the lattice of subspaces of an $\mathbb{F}_q$-linear space, say, $X$, generated by a set of projective points $A \subseteq X$. In this context, we introduce the notion of subspace distribution and show that partial knowledge of the latter is equivalent to partial knowledge of the Whitney numbers of $\mathcal{A}$. This refines a classical result by Dowling. The most interesting applications of our results are to be seen in the theory of higher-weight Dowling lattices (HWDLs), to which we devote the second and most substantive part of the paper. These combinatorial geometries were introduced by Dowling in 1971 in connection with fundamental problems in coding theory, most notably the famous MDS conjecture. They were further studied by, among others, Zaslavsky, Bonin, Kung, Brini, and Games. To date, still very little is known about these lattices and the techniques to compute their Whitney numbers have not been discovered yet. In this paper, we bring forward the theory of HWDLs, computing their Whitney numbers for new infinite families of parameters. We also show that the second Whitney numbers of HWDLs are polynomials in the underlying field size $q$, whose coefficients are curious expressions involving the Bernoulli numbers. In passing, we obtain new results intersecting coding theory and enumerative combinatorics.