denotes the number of partitions of into at most parts. See Table 26.9.1.
Unrestricted partitions are covered in §27.14.
A useful representation for a partition is the Ferrers graph in which the integers in the partition are each represented by a row of dots. An example is provided in Figure 26.9.1.
The conjugate partition is obtained by reflecting the Ferrers graph across the main diagonal or, equivalently, by representing each integer by a column of dots. The conjugate to the example in Figure 26.9.1 is . Conjugation establishes a one-to-one correspondence between partitions of into at most parts and partitions of into parts with largest part less than or equal to . It follows that also equals the number of partitions of into parts that are less than or equal to .
is the number of partitions of into at most parts, each less than or equal to . It is also equal to the number of lattice paths from to that have exactly vertices , , , above and to the left of the lattice path. See Figure 26.9.2.
In what follows
equivalently, partitions into at most parts either have exactly parts, in which case we can subtract one from each part, or they have strictly fewer than parts.
where the inner sum is taken over all positive divisors of that are less than or equal to .
As with fixed,