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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
2Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA
Annals of Combinatorics 13 (1) pp.1-25 March, 2009
AMS Subject Classification: 05A30; 05A19, 05A15, 05A05, 05E15
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


1. R.M. Adin, F. Brenti, and Y. Roichman, Descent numbers and major indices for the hyperoctahedral group, Adv. Appl. Math. 27 (2-3) (2001) 210-–224.

2. R.M. Adin and Y. Roichman, The flag major index and group actions on polynomial rings, European J. Combin. 22 (4) (2001) 431-–446.

3. C.-O. Chow, On derangement polynomials of type B, S´em. Lothar. Combin. 55 (2006) Article B55b.

4. A.M. Garsia and J.B. Remmel, A combinatorial interpretation of q-derangement and q- Laguerre numbers, European J. Combin. 1 (1980) (1) 47-–59.

5. P.A. MacMahon, Combinatory Analysis, Vol. I, II, Cambridge University Press, Cambridge, 1915, 1916.

6. J.B. Remmel, A Note on a Recursion for the Number of Derangements, European J. Combin. 4 (4) (1983) 371-–374.

7. M.L.Wachs, On q-derangement numbers, Proc. Amer. Math Soc. 106 (1) (1989) 273–-278.