20
CHU Wenchang and DI CLAUDIO Leontina
It is obvious that µ is a partition of |λ| − (1+m
2 ) into ≤ m parts. As an example, the following figures show this correspondence between two partitions
λ = (97431) and µ = (4311).
λ = (97431)
µ = (4311)
It is not difficult to verify that the mapping (B3.2) is a bijection between the
partitions of n with exactly m distinct parts and the partitions of n − (1+m
2 )
with ≤ m parts. Therefore
the
generating
function
of
{Q
(n)}
is
equal
to
m
n
1+m
that of {pm n − (1+m
,
the
number
of
partitions
of
n
−
}
with
the
(
)
)
n
2
2
number of parts ≤ m:
∞
X
Qm (n) qn =
n=0
= q(
∞
X
n
pm n − (1+m
2 ) q
n=0
1+m
2
)
∞
X
n=0
pm (n) qn =
1+m
q( 2 )
(q; q)m
thanks for the generating function displayed in (B2.4). This completes the
proof of Theorem B3.1.
Instead of the ordinary Ferrers diagram, we can draw a shifted diagram of
λ as follows (see the figure). Under the first row of λ1 squares, we put λ2
squares lined up vertically from the second column. For the third row, we