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Zaslavsky, Thomas. (1403). Whitney numbers of partial dowling lattices. , 13(2), 169-178. doi: 10.22108/toc.2023.135127.2013
Thomas Zaslavsky. "Whitney numbers of partial dowling lattices". , 13, 2, 1403, 169-178. doi: 10.22108/toc.2023.135127.2013
Zaslavsky, Thomas. (1403). 'Whitney numbers of partial dowling lattices', , 13(2), pp. 169-178. doi: 10.22108/toc.2023.135127.2013
Zaslavsky, Thomas. Whitney numbers of partial dowling lattices. , 1403; 13(2): 169-178. doi: 10.22108/toc.2023.135127.2013

Whitney numbers of partial dowling lattices

Transactions on Combinatorics
مقاله 7، دوره 13، شماره 2، شهریور 2024، صفحه 169-178    اصل مقاله (452.14 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/toc.2023.135127.2013
نویسنده
Thomas Zaslavsky*
Binghamton University, Binghamton, NY 13902-6000, U.S.A.
چکیده
The Dowling lattice $Q_n(G)$, $G$ a finite group, generalizes the geometric lattice generated by all vectors, over a field, with at most two nonzero components. Abstractly, it is a fundamental object in the classification of finite matroids. Constructively, it is the frame matroid of a certain gain graph known as $G K{_n}{^V}$. Its Whitney numbers of the first kind enter into several important formulas. Ravagnani suggested and partially proved that these numbers of $Q_n(G)$ and higher-weight generalizations are polynomial functions of $|G|$. We give a simple proof for $Q_n(G)$ and its generalization to a wider class of gain graphs and biased graphs, and we determine the degrees and coefficients of the polynomials.
کلیدواژه‌ها
Matroid؛ Whitney number؛ Dowling lattice؛ group expansion gain graph
مراجع

[1] T. A. Dowling, A class of geometric lattices based on finite groups, J. Combin. Theory Ser. B, 14 (1973) 61–86.
[2] J. Kahn and J. P. S. Kung, Varieties of combinatorial geometries, Trans. Amer. Math. Soc., 271 (1982) 485–499.
[3] A. Ravagnani, Whitney numbers of combinatorial geometries and higher-weight Dowling lattices, SIAM J. Appl.
Algebra Geom., 6 (2022) 156–189.
[4] G.-C. Rota, On the foundations of combinatorial theory: I. Theory of Möbius functions. Z. Wahrsch. verw. Gebiete,
2 (1964) 340–368.
[5] H. Whitney, A logical expansion in mathematics, Bull. Amer. Math. Soc., 38 (1932) 572–579.
[6] T. Zaslavsky, Biased graphs. II. The three matroids, J. Combin. Theory Ser. B, 51 (1991) 46–72.
[7] T. Zaslavsky, Biased graphs. III. Chromatic and dichromatic invariants, J. Combin. Theory Ser. B, 64 (1995) 17–88.
[8] T. Zaslavsky, Associativity in multiary quasigroups: The way of biased expansions, Aequat. Math., 83 (2012) 1–66.

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