Partition theory is a branch of number theory that studies the ways in which a positive integer can be expressed as the sum of positive integers, disregarding the order of the addends. For example, the number 4 can be partitioned into 4, 3+1, 2+2, and 2+1+1, resulting in five distinct partitions. The study of partitions has connections to various areas of mathematics, including combinatorics, q-series, and mathematical analysis. Notable mathematicians like Srinivasa Ramanujan made significant contributions to partition theory, leading to deeper insights and formulas that describe the number of partitions for large integers.