According to whether or not the arrangement order between partition parts is considered, we can divide the integer partition problem into ordered combination and disorderly partition. The differences between the two are as follows:
In an orderly split, consider the order between the split parts and. suppose
Different sorting schemes are called different schemes, which are called ordered K-splitting of n. For example, the ordered 2-splitting of 3 is: 3= 1+2=2+ 1. We can model this problem as a "partition" problem in permutation and combination, that is, if N undifferentiated balls are divided into R parts, and each part has at least one ball, r- 1 partition needs to be inserted into the gap between n- 1 balls, so the total number of * * schemes is C (N- 1.
In disorderly division, the order of summation is not considered. General assumption
We call it disordered K-splitting of n, for example, disordered K-splitting of 3 is: 3= 1+2. This splitting can be understood as dividing n undifferentiated balls into r parts, each part has at least one ball.
Generally speaking, the number of disorderly splits is represented by p(n), so p(2)= 1, p(3)=2, and p(4)=4.
Generally speaking, integer splitting refers to the disorderly splitting of integers.
Integer partition theory mainly studies the properties and relationships of various partition functions. As early as the Middle Ages, there was research on special integer partition. In the 1940s, L. Euler put forward the generating function method (or formal power series method) to study integer partition, and proved many important theorems, which laid a theoretical foundation for integer partition. The introduction of circle method in analytic number theory further develops the theory of integer partition. Integer partition is closely related to modular function and has important applications in combinatorial mathematics, group theory, probability theory, mathematical statistics and particle physics.