24
CHU Wenchang and DI CLAUDIO Leontina
B4. Partitions and the Gauss q-binomial coefficients
B4.1. Lemma. Let p` (n|m) and p` (n|m) be the numbers of partitions of
n into ` and ≤ ` parts, respectively, with each part ≤ m. We have the
generating functions:
X
p` (n|m) x`qn
=
1
(1 − xq)(1 − xq2 ) · · · (1 − xqm )
(B4.1a)
p` (n|m) x`qn
=
1
.
(1 − x)(1 − xq) · · · (1 − xqm )
(B4.1b)
`,n≥0
X
`,n≥0
The first identity (B4.1a) is a special case of the generating function shown
in (B1.1b).
On account of the length of partitions, we have
p` (n|m) = p0 (n|m) + p1 (n|m) + · · · + p` (n|m).
Manipulating the triple sum and then applying the geometric series, we can
calculate the corresponding generating function as follows:
X
p`(n|m) x`qn
=
X X̀
pk (n|m)x` qn
=
`,n≥0 k=0
∞
∞ X
X
∞
X
=
k=0 n=0
∞
∞ X
X
`,n≥0
pk (n|m)qn
1
1−x
x`
`=k
pk (n|m)xk qn .
k=0 n=0
The last expression leads us immediately to the second bivariate generating
function (B4.1b) in view of the first generating function (B4.1a).
B4.2. The Gauss q-binomial coefficients as generating functions.
Let p` (n|m) and p` (n|m) be as in Lemma B4.1. The corresponding univariate generating functions read respectively as
X
`+m−1 `
n
p` (n|m) q
=
q
(B4.2a)
m−1
n≥0
X
`+m
`
n
p (n|m) q
=
(B4.2b)
m
n≥0