# Finding a specific case for the partitions of $n$

A generating function for the total number of partitions of $$n$$ is: $$\prod_{i=1}^n\sum_{j=0}^n x^{ij}$$ The polynomial generated by this generating function will have some term $$x^n$$, the coefficient of which will equal the total number of partitions of $$n$$.

Let $$n$$=6. The eleven partitions of 6 are the following: 6, 5+1, 4+2, 3+3, 4+1+1, 3+2+1, 2+2+2, 3+1+1+1, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1+1. I have set created two cases.

Case 1: where $$k\le3$$ and the maximum value of a term in a partition is 4 (where 4 is equal to $$\frac{2n}{k}$$). The partitions of 6 that follow this are: 4+2, 3+3*, 4+1+1*, 3+2+1, 2+2+2.

Case 2: where $$k\le4$$ and the maximum value of a term in a partition is 3 (where 3 is equal to $$\frac{2n}{k}$$). The partitions of 6 that follow this are: 3+3, 3+2+1, 2+2+2*, 3+1+1+1*, 2+2+1+1.

I am also investigating the specific partitions (noted with a "*") where every term in the partition subtracted from $$\frac{2n}{k}$$ creates another partition of $$n$$. For example, in case 1, where $$\frac{2n}{k}=4$$, subtracting the terms 4, 1, and 1 (the partition 4+1+1) from 4 gives 0, 3 and 3 (the partition 3+3). Also in case 2 where $$\frac{2n}{k}=3$$, subtracting the terms 3, 1, 1, and 1 (3+1+1+1) from 3 gives 0, 2, 2 and 2 (the partition 2+2+2). The other partitions create themselves when each term is subtracted from $$\frac{2n}{k}$$.

What generating function can be used to

1) find the number of partitions that satisfy the parameters where $$k\le$$ some number?

2) find the number of partitions where maximum value for a term in a partition does not exceed $$\frac{2n}{k}$$?

3) find the number of partitions that, when the terms are subtracted from $$\frac{2n}{k}$$, create a different partition of $$n$$?

I am very new to combinatorics and would appreciate dumbed-down answers. Thank you!