proofs

Identities versus bijections 211
As experienced formal series manipulators we notice that the introduction
of the new variable y yields
∏
(1 + yx k ) = ∑
p d,m (n)x n y m ,
k≥1
n,m≥0
where p d,m (n) counts the partitions of n into precisely m distinct summands.
With y = −1 this yields
∏
− x
k≥1(1 k ) = ∑ (E d (n) − O d (n))x n , (10)
n≥0
where E d (n) is the number of partitions of n into an even number of distinct
parts, and O d (n) is the number of partitions into an odd number. And here
is the punchline. Comparing (10) to Euler’s expansion in (8) we infer the
beautiful result
⎧
1 for n =
⎪⎨
3j2 ±j
2
when j ≥ 0 is even,
E d (n) − O d (n) = −1 for n = 3j2 ±j
2
when j ≥ 1 is odd,
⎪⎩
0 otherwise.
An example for n = 10:
10 = 9 + 1
10 = 8 + 2
10 = 7 + 3
10 = 6 + 4
10 = 4 + 3 + 2 + 1
and
10 = 10
10 = 7 + 2 + 1
10 = 6 + 3 + 1
10 = 5 + 4 + 1
10 = 5 + 3 + 2,
so E d (10) = O d (10) = 5.
This is, of course, just the beginning of a longer and still ongoing story. The
theory of infinite products is replete with unexpected indentities, and with
their bijective counterparts. The most famous examples are the so-called
Rogers–Ramanujan identities, named after Leonard Rogers and Srinivasa
Ramanujan, in which the number 5 plays a mysterious role:
∏
k≥1
1
(1 − x 5k−4 )(1 − x 5k−1 )
= ∑ n≥0
x n2
(1 − x)(1 − x 2 ) · · · (1 − x n ) ,
∏
k≥1
1
(1 − x 5k−3 )(1 − x 5k−2 )
= ∑ n≥0
x n2 +n
(1 − x)(1 − x 2 ) · · · (1 − x n ) .
The reader is invited to translate them into the following partition identities
first noted by Percy MacMahon:
Srinivasa Ramanujan
• Let f(n) be the number of partitions of n all of whose summands are
of the form 5k + 1 or 5k + 4, and g(n) the number of partitions whose
summands differ by at least 2. Then f(n) = g(n).
• Let r(n) be the number of partitions of n all of whose summands are
of the form 5k + 2 or 5k + 3, and s(n) the number of partitions whose
parts differ by at least 2 and which do not contain 1. Then r(n) = s(n).
All known formal series proofs of the Rogers–Ramanujan identities are
quite involved, and for a long time bijection proofs of f(n) = g(n) and
of r(n) = s(n) seemed elusive. Such proofs were eventually given 1981
by Adriano Garsia and Stephen Milne. Their bijections are, however, very
complicated — Book proofs are not yet in sight.