## Samenvatting

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 M\" obius functions of nested atomistic lattices, extending some classical theorems in combinatorics. We then specialize our results to restriction geometries, i.e., to sublattices \scrA of the lattice of subspaces of an \BbbF 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 \scrA . 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.

Originele taal-2 | Engels |
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Pagina's (van-tot) | 156-189 |

Aantal pagina's | 34 |

Tijdschrift | SIAM Journal on Applied Algebra and Geometry |

Volume | 6 |

Nummer van het tijdschrift | 2 |

DOI's | |

Status | Gepubliceerd - 2022 |

### Bibliografische nota

Publisher Copyright:© 2022 Society for Industrial and Applied Mathematics

### Financiering

\ast Received by the editors November 25, 2020; accepted for publication (in revised form) September 21, 2021; published electronically April 13, 2022. https://doi.org/10.1137/20M1382635 Funding: The author was partially supported by the Marie Curie Research Grants Scheme through grant 740880 and by the Dutch Research Council through grant VI.Vidi.203.045. \dagger Department of Mathematics and Computer Science, Eindhoven University of Technology, 5612 AZ, Eindhoven, The Netherlands ([email protected]).

Financiers | Financiernummer |
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H2020 Marie Skłodowska-Curie Actions | |

European Union’s Horizon Europe research and innovation programme | 740880 |

Nederlandse Organisatie voor Wetenschappelijk Onderzoek | VI.Vidi.203.045 |