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A p,q-Analogue of the Generalized Derangement Numbers
Karen S. Briggs1 and Jeffrey B. Remmel2
1Department of Mathematics and Computer Science, North Georgia College & State University, Dahlonega, GA 30597, USA
kbriggs@ngcsu.edu
2Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA
jremmel@ucsd.edu
Annals of Combinatorics 13 (1) pp.1-25 March, 2009
AMS Subject Classification: 05A30; 05A19, 05A15, 05A05, 05E15
Abstract:
In this paper, we study the numbers $D_{n,\,k}$ which are defined as the number of permutations $\sg$ of the symmetric group $S_n$ such that $\sg$ has no cycles of length $j$ for $j \leq k$. In the case $k=1$, $D_{n,\,1}$ is simply the number of derangements of an $n$-element set. As such, we shall call the numbers $D_{n,\,k}$ generalized derangement numbers. Garsia and Remmel \cite{GR} defined some natural $q$-analogues of $D_{n,\,1}$, denoted by $D_{n,\,1}(q)$, which give rise to natural $q$-analogues of the two classical recursions of the number of derangements. The method of Garsia and Remmel can be easily extended to give natural $p,\,q$-analogues $D_{n,\,1}(p,\,q)$ which satisfy natural $p,\,q$-analogues of the two classical recursions for the number of derangements. In \cite{GR}, Garsia and Remmel also suggested an approach to define $q$-analogues of the numbers $D_{n,\,k}$. In this paper, we show that their ideas can be extended to give a $p,\,q$-analogue of the generalized derangements numbers. Again there are two classical recursions for generalized derangement numbers. However, the $p,\,q$-analogues of the two classical recursions are not as straightforward when $k \geq 2$.
Keywords: permutations, derangements, p,q-analogues

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