A p,q-Analogue of the
Generalized Derangement Numbers

Karen S. Briggs^{1} and Jeffrey B. Remmel^{2}
^{1}Department of
Mathematics and Computer Science, North Georgia College
& State University, Dahlonega, GA 30597, USA
^{2}Department of Mathematics, University of California, San Diego,
La Jolla, CA 92093, USA
**AMS Subject Classification: **05A30; 05A19, 05A15,
05A05, 05E15
**Keywords: **permutations, derangements, p,q-analogues

kbriggs@ngcsu.edu

jremmel@ucsd.edu

Annals of Combinatorics 13 (1) pp.1-25 March, 2009

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 $k \geq 2$.
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