Classical Partition Identities and Basic Hypergeometric Series
21
put λ3 squares beginning from the third column. Continuing in this way,
the last row of λm squares will be lined up vertically from the m-th column.
The shifted diagram
of partition λ = (97431)
From the shifted diagram of λ, we see that all the partitions enumerated
by Qm (n) have one common triangle on the left whose weight is (1+m
2 ). The
remaining parts right to the triangle are partitions of n − (m+1
with
≤m
)
2
parts. This reduces the problem of computing the generating function to
the case just explained.
B3.2. Classifying all the partitions with distinct parts according to the
number of parts, we get Euler’s classical partition identity
∞
Y
n
(1 − xq ) = 1 +
n=0
∞
X
m=1
m
(−1)m xm q( 2 )
(1 − q)(1 − q2 ) · · · (1 − qm )
(B3.3)
which can also be verified through the correspondence between partitions
into distinct odd parts and self-conjugate partitions.
Proof. Considering the bivariate generating function of Qm (n), we have
∞
Y
(1 + xqk ) =
k=1
∞
X
m=0
xm
∞
X
n=0
Qm (n)qn .