UC San Diego

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.