
Generalizations of Stirling Numbers 119
Similarly, the numbers
˜
S
2,p,q
n,k
(α, β, r) satisfy
[t|β]
n|p,q
=
n
k=0
˜
S
2,p,q
n,k
(α, β, r)[t −r|α]
k|p,q
.
Remmel and Wachs also gave an interpretation for these generalized Stirling numbers (for
particular choices of parameters) in terms of rooks; see [928]. We want to quote two other
results from their paper. For this, we need to introduce some notation from [928]. In the
following, we set β = 0, and we will consider α = j and r = i to be integers. We also replace
t by x + i, and we will use the recurrence relation
˜
S
2,p,q
n+1,k
(j, 0,i)=q
i+(k−1)j
˜
S
2,p,q
n,k−1
(j, 0,i)+p
x−kj
[kj + i]
p,q
˜
S
2,p,q
n,k
(j, 0,i).
Let us, furthermore, ...