A p , q -Analogue of the Generalized Derangement Numbers

In this paper, we study the numbers D n , k which are defined as the number of permutations σ of the symmetric group S n such that σ has no cycles of length j for j ≤ 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 [4] 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 [4], 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 eneralized derangement numbers. However, the p , q -analogues of the two classical recursions are not as straightforward when k ≥ 2.