
94 Commutation Relations, Normal Ordering, and Stirling Numbers
Theorem 4.35 (M´endez, Rodr´ıguez) For al l s
1
≤ k ≤|s|, one has that
S
r,s|q
(k)=
(−1)
k
k!
k
p=s
1
(−1)
p
q
(
k−p
2
)
k
p
q
n
m=1
[d
m−1
+ p]
s
m
|q
. (4.41)
In the uniform case, (4.41) reduces to (4.30) for r = s .
M´endez and Rodr´ıguez [808, 809] gave an interpretation in terms of colonies and bugs,
where certain q-weights are introduced. In addition to the terminology introduced above,
we call a bug of type (r, s)aworm if s = 0 (that is, it has no legs). When constructing a
colony, after placing all the bugs, a ghost bug is placed. Its legs do not cross among them
and occupy all the empty cells of the