Classical Partition Identities and Basic Hypergeometric Series
17
A combinatorial proof. Let p(n|λ1 = m) be the number of partitions of
n with the first part λ1 equal to m. Then pm (n) = p(n|λ1 = m) because
the partitions enumerated by pm (n) are conjugate with those enumerated
by p(n|λ1 = m). Therefore they have the same generating functions:
∞
X
pm (n) q
n
n=0
=
∞
X
p(n|λ1 = m) qn.
n=0
All the partition of n enumerated by p(n|λ1 = m) have the first part λ1 = m
in common and the remaining parts constitute the partitions of n − m with
each part ≤ m. Therefore we have
∞
X
p(n|λ1 = m) qn
=
∞
X
=
n=m
∞
X
m
n=0
p(n − m|λ1 ≤ m) qn = qm
q
∞
X
p(n|λ1 ≤ m) qn
n=0
p(n|{1, 2, · · · , m}) qn =
n=0
qm
(q; q)m
where the first line is justified by replacement n → n + m on summation
index, while the second is a consequence of (B1.1a).
This confirms again the generating function (B2.1).
B2.2. Proposition. Let pm (n) be the number of partitions into ≤ m parts
(or dually, partitions into parts ≤ m). Then we have the generating function
∞
X
n=0
pm (n) qn =
1
(1 − q)(1 − q2) · · · (1 − qm )
(B2.4)
which yields a finite summation formula
m
X
1
qk
=
1
+
.
(1 − q)(1 − q2 ) · · · (1 − qm )
(1 − q)(1 − q2 ) · · · (1 − qk )
k=1
Proof. Notice that pm (n), the number of partitions into ≤ m parts is equal
to the number of partitions into parts ≤ m in view of conjugate partitions.
We get immediately from (B1.1a) the generating function (B2.4).
The classification of the partitions of n into ≤ m parts with respect to the
number k of parts yields
pm (n) = p0(n) + p1 (n) + p2(n) + · · · + pm (n).