Geodesic
The Homology of the Real Complement of a $k$-parabolic Subspace Arrangement
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010).

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The $k$-parabolic subspace arrangement, introduced by Barcelo, Severs and White, is a generalization of the well known $k$-equal arrangements of type-$A$ and type-$B$. In this paper we use the discrete Morse theory of Forman to study the homology of the complements of $k$-parabolic subspace arrangements. In doing so, we recover some known results of Björner et al. and provide a combinatorial interpretation of the Betti numbers for any $k$-parabolic subspace arrangement. The paper provides results for any $k$-parabolic subspace arrangement, however we also include an extended example of our methods applied to the $k$-equal arrangements of type-$A$ and type-$B$. In these cases, we obtain new formulas for the Betti numbers. 
@article{DMTCS_2010_special_259_a77,
     author = {Severs, Christopher and White, Jacob A.},
     title = {The {Homology} of the {Real} {Complement} of a $k$-parabolic {Subspace} {Arrangement}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)},
     year = {2010},
     doi = {10.46298/dmtcs.2882},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2882/}
}
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%A White, Jacob A.
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Severs, Christopher; White, Jacob A. The Homology of the Real Complement of a $k$-parabolic Subspace Arrangement. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010). doi : 10.46298/dmtcs.2882. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2882/

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