Classical Partition Identities and Basic Hypergeometric Series
13
in which the coefficient of x` qn reads as
Y
X P
P
1
` n
k∈S mk q
k∈S kmk
[x`qn ]
=
[x
q
]
x
1 − xqk
k∈S
mk ≥0
k∈S
=
X
P
o
Pk∈S mk = `
k∈S kmk = n
1.
mk ≥0: k∈S
The last sum enumerates the solutions of the system of Diophantine equations
X
mk = `
k∈S
X
kmk = n
k∈S
which are the number of partitions 1m1 , 2m2 , · · · , nmn of n with exactly
`-parts in S.
B1.2. Partitions into distinct parts in S. Next we study the generating
functions of partitions into distinct parts in S.
If we denote by Q(n|S) and Q` (n|S) the corresponding partition numbers
with distinct parts from S, then their generating functions read respectively
as
∞
X
Y
1 + qk
(B1.2a)
Q(n|S) qn
=
n=0
X
k∈S
` n
Q` (n|S) x q
=
`,n≥0
Y
1 + xqk .
(B1.2b)
k∈S
Proof. For the first identity, observing that
X
1 + qk =
qkmk
mk =0,1
we can reformulate the product on the right hand side as
Y
Y X
X P
1 + qk
=
qkmk =
q k∈S kmk .
k∈S mk =0,1
k∈S
mk =0,1
k∈S
Extracting the coefficient of qn , we obtain
Y
X P
[qn]
1 + qk = [qn]
q k∈S kmk =
k∈S
mk =0,1
k∈S
P
X
k∈S kmk =n
mk =0,1: k∈S
1.