
A Generalization of the Weyl Algebra 317
via the binomial identity (see, for instance, [508])
n − 1
k − 1
=
k
j=0
(−1)
k−j
k
j
n + j − 1
j
.
Let L(n)=
n
k=0
L(n, k)andL(n)=∪
n
k=0
L(n, k). Then L(n)=|L(n)|, the cardinality of
the set of all distributions of n labeled balls in unlabeled, contents-ordered boxes. The L(n)
are analogs of the usual Bell numbers; see, for example, [848], where they are described as
counting sets of lists having size n. Letting s =1/2, h = 2 in Corollary 8.89, we obtain a
Dobi´nski-type formula for L(n), namely
L(n)=
1
e
j≥0
j
n
j!
. (8.87)
We now proceed in supplying a combinatorial interpretation for the numbers S
a;b
(n, k)
defined by