When you interpret the elements of a group as a linear map (aka matrix), you can apply certain tools from linear algebra to study groups. This is known as representation theory.

One famous result from representation theory is that the number of inequivalent irreducible representations of the symmetric group is equal to the number of partitions of n.

In general, the number of irreducible representations of a group G is equal to the number of conjugacy classes of G.

So, if we take S_5, the conjugacy classes correspond to cycle types (abcde), (abcd), (abc)(de), (ab)(cd), (ab), and e. Hence there are 7 inequivalent irreducible representations of S_5. One can check that there are also 7 partitions of 5: 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1.