In number theory, Glaisher's theorem is an identity useful to the study of integer partitions. Proved in 1883 [1] by James Whitbread Lee Glaisher, it states that the number of partitions of an integer into parts not divisible by is equal to the number of partitions in which no part is repeated or more times. This generalizes a result established in 1748 by Leonhard Euler for the case .
It states that the number of partitions of an integer into parts not divisible by is equal to the number of partitions in which no part is repeated d or more times, which can be written formally as partitions of the form where and .
When this becomes the special case known as Euler's theorem, that the number of partitions of into distinct parts is equal to the number of partitions of into odd parts.
In the following examples, we use the multiplicity notation of partitions. For example, is a notation for the partition 1 + 1 + 1 + 1 + 2 + 3 + 3.
Among the 15 partitions of the number 7, there are 5, shown in bold below, that contain only odd parts (i.e. only odd numbers):
If we count now the partitions of 7 with distinct parts (i.e. where no number is repeated), we also obtain 5:
It should be noted that the partitions in bold in the first and second case are not the same, and it is not obvious why their number is the same.
Among the 11 partitions of the number 6, there are 7, shown in bold below, that contain only parts not divisible by 3:
And if we count the partitions of 6 with no part that repeats more than 2 times, we also obtain 7:
A proof of the theorem can be obtained with generating functions. If we note the number of partitions with no parts divisible by d and the number of partitions with no parts repeated more than d-1 times, then the theorem means that for all n . The uniqueness of ordinary generating functions implies that instead of proving that for all n, it suffices to prove that the generating functions of and are equal, i.e. that .
Each generating function can be rewritten as infinite products (with a method similar to the infinite product of the partition function) :
If we expand the infinite product for :
we see that each term in the numerator cancels with the corresponding multiple of d in the denominator. What remains after canceling all the numerator terms is exactly the infinite product for .
Hence the generating functions for and are equal.
If instead of counting the number of partitions with distinct parts we count the number of partitions with parts differing by at least 2, a further generalization is possible. It was discovered by Leonard James Rogers, and then independently by Schur and Ramanujan, in what are now known as the Rogers-Ramanujan identities. It states that:
For example, there are only 3 partitions of 7, shown in bold below, into parts differing by at least 2 (note: if a number is repeated in a partition, it means a difference of 0 between two parts, hence the partition is not counted):
And there are also only 3 partitions of 7 involving only the parts 1, 4, 6:
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