22
CHU Wenchang and DI CLAUDIO Leontina
Recalling (B3.1) and then noting that Q0(n) = δ0,n, we deduce that
∞
Y
(1 + xqk ) = 1 +
k=1
1+m
∞
X
xm q ( 2 )
(q; q)m
m=1
which becomes the Euler identity under parameter replacement x → −x/q.
In view of Euler’s Theorem A2.1, we have a bijection between the partitions
into distinct odd parts and the self-conjugate partitions.
The self-conjugate partition
λ = (653221)
with the Durfee square 3 × 3
For a self-conjugate partition with the main diagonal length equal to m
(which corresponds exactly to the length of partitions into distinct odd
parts), it consists of three pieces: the first piece is the square of m × m
2
on the top-left with bivariate enumerator xm qm , the second piece right to
the square is a partition with ≤ m parts enumerated by 1/(q; q)m and the
third piece under the square is in effect the conjugate of the second one.
Therefore the partitions right to the square and under the square m × m
are altogether enumerated by 1/(q2; q2)m .
Classifying the self-conjugate partitions according to the main diagonal
length m, multiplying both generating functions together and summing m