Whitney numbers of partial dowling lattices

Zaslavsky, Thomas

doi:10.22108/toc.2023.135127.2013

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	Whitney numbers of partial dowling lattices
Transactions on Combinatorics
مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 07 خرداد 1402 اصل مقاله (464.45 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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Whitney numbers of partial dowling lattices