Is there any direct application of all developments related to partitions? I am specially interested in physics but cryptography or other mostly theoretical areas would also be a good answer.
By partitions I mean all different ways to write down an integer as sum of integers (not necesary to be differents) and in which the order in all its parts do no matter. For instance these are partitions of 5:
5 = 5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1
so p(5) = 7. There are a lot of refinements and variations of the concept but all have in common sum of parts with order meaningless.
Not a "theoretical area" per se, but IIRC, ATMs use partitions of integers to detemerine the number and denominations of currency notes that must be used to achieve the sum (of money) requested by the customer.