# Use the principle of inclusion-exclusion to find the number of integer partitions of n in which exactly one of 4,7 and 13 is a part

Use the principle of inclusion-exclusion to find the number of integer partitions of $$n$$ in which exactly one of 4,7 and 13 is a part. The answer can be written as a linear combination of terms $$p(k)$$ for various $$k$$.

So I've started this by setting the ambient set as $$A=[n]$$. And defining the three properties as

$$p_1$$: 4 is a part of the partition,
$$p_2$$: 7 is a part of the partition, and
$$p_3$$: 13 is a part of the partition.

I have the formula $$e(X)=\sum_{Y:X\subseteq Y \subseteq P} (-1)^{|Y-X|}a(Y)$$
With $$e(X)=$$ the number of elements in A with exactly the properties in X
and $$a(X)=$$ the number of elements in A with at least the properties in X.

I know that I need to find $$e(p_1)+e(p_2)+e(p_3)$$, I'm just not sure how to do this

I will use $$a(p_1,p_2)$$ to denote the number of partitions containing $$4$$ and $$7$$, and similarly $$a(p_1,p_2,p_3)$$ counts partitions which contain $$4,7$$ and $$13$$. Using your formula,
$$e(p_1)=a(p_1)-a(p_1,p_2)-a(p_1,p_3)+a(p_1,p_2,p_3).\tag1$$
You can get similar expressions for $$e(p_2)$$ and $$e(p_3)$$. Furthermore,
• $$a(p_1)=p(n-4)$$, $$a(p_2)=p(n-7)$$, etc, because a partition containing $$4$$ corresponds exactly to a partition of $$n-4$$ with a part of $$4$$ added.
• $$a(p_1,p_2)=p(n-4-7)$$, etc, for the same reason.
• $$a(p_1,p_2,p_3)=p(n-4-7-13)$$.
Now, you just need to plug these expressions for $$a(\dots)$$ into the expressions for $$e(p_i)$$ in $$(1)$$.