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On Whitney numbers of Dowling lattices. (English) Zbl 0861.05004
Let $(L,\le)$ be a finite lattice and let $h(x)$ be the height function on $L$. The Whitney numbers of the second kind $W(L,k)$ for $L$ are deferred by $W(L,k)=|\{x\in L:h(x)=k\}|$. Let $W_m(n,k)$ denote the Withney numbers for the Dowling geometric lattice $Q_n(G)$ over a group $G$ of order $m$. From the author’s introduction: The organization of the paper is as follows. In Section 2, we determine the generating function for the sequence $W_m(n,k)$, $0\le k\le n$. As a consequence, we derive an explicit formula, which is a generalization of the formula for the Stirling numbers of the second kind. Also, we consider the sum of Whitney numbers of the second kind, $D_m(n)$, which we call Dowling numbers; these numbers are a generalization of Bell’s. The generating function of $D_m(n)$ is used to locate with high precision the mode of the sequence $W_m(n,k)$ (Section 4). The third section is devoted to the Whitney numbers of the first kind. We give generating functions, as well as some recurrence and congruence relations. Those are essentially generalization of some facts known about Stirling numbers of the first kind. In the last section, we locate the modes of the Whitney numbers of both kinds. For the Whitney numbers of the second kind, the calculations are long and tedious, therefore we give only the idea and the results.

MSC:
05A15Exact enumeration problems, generating functions
06C10Semimodular lattices, geometric lattices
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References:
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