Q-counting rook configurations and a formula of Frobenius
Recurrences \(S_{n+1,k}(q)=S_{n,k-1}q^{k-1}+(1+q+...+q^{k- 1})S_{n,k}(q)\) for \(0\leq k\leq n\) with the initial conditions \(S_{0,0}(q)=1\) and \(S_{n,k}(q)=0\) if \(k<0\) or \(k>n\) define the polynomials \(S_{n,k}(q)\). To these polynomials \textit{S. Milne} [Adv. Math. 43, 173-196 (1982; Zbl 0482.05012)] gave a combinatorial interpretation in terms of partitions. In this paper the authors give a new combinatorial interpretation to \(S_{n,k}(q)\) as polynomials, q-counting the configurations of n-k non-taking rooks in some special Ferrer's board. This leads them to a new proof (via an interesting combinatorial interpretation) of the identity \[ \frac{\sum^{n}_{k=1}S_{n,k}(q)\prod^{k}_{i=1}(1+q+...+q^{i-1})x^ k}{(1- x)(1-xq)...(1-xq^ k)}=\sum_{k\geq 1}(1+q+...+q^{k-1})^ nx^ k, \] proved earlier by the first author [J. Linear Multilinear Algebra 8, 21- 34 (1980; Zbl 0442.05002)] by algebraic means. This identity may be viewed as q-analogue of an identity of Frobenius relating the Stirling numbers of second kind to Eulerian polynomials, to which it reduces for \(q=1\). A number of interesting questions about the polynomials \(S_{n,k}(q)\) are posed, e.g. the unimodality of their coefficients is conjectured. The authors show also their approach to be suitable for q-Laguerre polynomials studied earlier by \textit{W. Hahn} [Math. Nachr. 2, 4-34 (1949; Zbl 0031.39001)] and by the authors [Eur. J. Comb. 1, 47-59 (1980; Zbl 0462.05012)]. The considerations in this paper are based on the study of rook polynomials by \textit{J. R. Goldman, J. T. Joichi} and \textit{D. E. White} [Proc. Am. Math. Soc. 52, 485-492 (1975; Zbl 0312.05002)] and by \textit{J. R. Goldman, J. T. Joichi, D. L. Reiner} and \textit{D. E. White} [SIAM J. Appl. Math. 31, 618-633 (1976; Zbl 0359.05005)] and may be viewed as a q-extension of this study. As a corollary the authors remark that two rook boards with identical rook numbers have also identical q-rook numbers; a question about a purely combinatorial derivation of this fact is raised.
q-analogue of Frobenius identity for Stirling numbers of second kind
scientific article; zbMATH DE number 3963835
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