In both combinatorics and number theory, families of partitions subject to various restrictions are often studied. This section surveys a few such restrictions.
Conjugate and self-conjugate partitions If we flip the diagram of the partition 6 + 4 + 3 + 1 along its
main diagonal, we obtain another partition of 14: By turning the rows into columns, we obtain the partition 4 + 3 + 3 + 2 + 1 + 1 of the number 14. Such partitions are said to be
conjugate of one another. In the case of the number 4, partitions 4 and 1 + 1 + 1 + 1 are conjugate pairs, and partitions 3 + 1 and 2 + 1 + 1 are conjugate of each other. Of particular interest are partitions, such as 2 + 2, which have themselves as conjugate. Such partitions are said to be
self-conjugate.
Claim: The number of self-conjugate partitions is the same as the number of partitions with distinct odd parts.
Proof (outline): The crucial observation is that every odd part can be "
folded" in the middle to form a self-conjugate diagram: One can then obtain a
bijection between the set of partitions with distinct odd parts and the set of self-conjugate partitions, as illustrated by the following example:
Odd parts and distinct parts Among the 22 partitions of the number 8, there are 6 that contain only
odd parts: • 7 + 1 • 5 + 3 • 5 + 1 + 1 + 1 • 3 + 3 + 1 + 1 • 3 + 1 + 1 + 1 + 1 + 1 • 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 Alternatively, we could count partitions in which no number occurs more than once. Such a partition is called a
partition with distinct parts. If we count the partitions of 8 with distinct parts, we also obtain 6: • 8 • 7 + 1 • 6 + 2 • 5 + 3 • 5 + 2 + 1 • 4 + 3 + 1 This is a general property. For each positive number, the number of partitions with odd parts equals the number of partitions with distinct parts, denoted by
q(
n). This result was proved by
Leonhard Euler in 1748 and later was generalized as
Glaisher's theorem. For every type of restricted partition there is a corresponding function for the number of partitions satisfying the given restriction. An important example is
q(
n) (partitions into distinct parts). The first few values of
q(
n) are (starting with
q(0)=1): :1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, ... . The
generating function for
q(
n) is given by :\sum_{n=0}^\infty q(n)x^n = \prod_{k=1}^\infty (1+x^k) = \prod_{k=1}^\infty \frac {1}{1-x^{2k-1}} . The
pentagonal number theorem gives a recurrence for
q: :
q(
k) =
ak +
q(
k − 1) +
q(
k − 2) −
q(
k − 5) −
q(
k − 7) +
q(
k − 12) +
q(
k − 15) −
q(
k − 22) − ... where
ak is (−1)
m if
k = 3
m2 −
m for some integer
m and is 0 otherwise.
Restricted part size or number of parts By taking conjugates, the number of partitions of into exactly
k parts is equal to the number of partitions of in which the largest part has size . The function satisfies the recurrence : with initial values and if and and are not both zero. One recovers the function
p(
n) by : p(n) = \sum_{k = 0}^n p_k(n). One possible generating function for such partitions, taking
k fixed and
n variable, is : \sum_{n \geq 0} p_k(n) x^n = x^k\prod_{i = 1}^k \frac{1}{1 - x^i}. More generally, if
T is a set of positive integers then the number of partitions of
n, all of whose parts belong to
T, has
generating function :\prod_{t \in T}(1-x^t)^{-1}. This can be used to solve
change-making problems (where the set
T specifies the available coins). As two particular cases, one has that the number of partitions of
n in which all parts are 1 or 2 (or, equivalently, the number of partitions of
n into 1 or 2 parts) is :\left \lfloor \frac{n}{2}+1 \right \rfloor , and the number of partitions of
n in which all parts are 1, 2 or 3 (or, equivalently, the number of partitions of
n into at most three parts) is the nearest integer to (
n + 3)2 / 12.
Partitions in a rectangle and Gaussian binomial coefficients One may also simultaneously limit the number and size of the parts. Let denote the number of partitions of with at most parts, each of size at most . Equivalently, these are the partitions whose Young diagram fits inside an rectangle. There is a recurrence relation p(N,M;n) = p(N,M-1;n) + p(N-1,M;n-M) obtained by observing that p(N,M;n) - p(N,M-1;n) counts the partitions of into exactly parts of size at most , and subtracting 1 from each part of such a partition yields a partition of into at most parts. The Gaussian binomial coefficient is defined as: {k+\ell \choose \ell}_q = {k+\ell \choose k}_q = \frac{\prod^{k+\ell}_{j=1}(1-q^j)}{\prod^{k}_{j=1}(1-q^j)\prod^{\ell}_{j=1}(1-q^j)}. The Gaussian binomial coefficient is related to the
generating function of by the equality \sum^{MN}_{n=0}p(N,M;n)q^n = {M+N \choose M}_q. ==Rank and Durfee square==