Question 13
A partition of a number $n$ is a sequence of positive integers, arranged in descending order, whose sum is $n$. For example, $n=4$ has 5 partitions: $1+1+1+1=2+1+1=2+2=3+1=4$. Given two different partitions of the same number, $n=a_{1}+a_{2}+\cdots+a_{k}=b_{1}+b_{2}+\cdots+b_{l}$, where $k \leq l$, the first partition is said to dominate the second if all of the following inequalities hold:
$$
\begin{aligned}
a_{1} & \geq b_{1} ; \\
a_{1}+a_{2} & \geq b_{1}+b_{2} ; \\
a_{1}+a_{2}+a_{3} & \geq b_{1}+b_{2}+b_{3} ; \\
& \vdots \\
a_{1}+a_{2}+\cdots+a_{k} & \geq b_{1}+b_{2}+\cdots+b_{k} .
\end{aligned}
$$
Find as many partitions of the number $n=20$ as possible such that none of the partitions dominates any other. Your score will be the number of partitions you find. If you make a mistake and one of your partitions does dominate another, your score is the largest $m$ such that the first $m$ partitions you list constitute a valid answer.
$$
\begin{aligned}
a_{1} & \geq b_{1} ; \\
a_{1}+a_{2} & \geq b_{1}+b_{2} ; \\
a_{1}+a_{2}+a_{3} & \geq b_{1}+b_{2}+b_{3} ; \\
& \vdots \\
a_{1}+a_{2}+\cdots+a_{k} & \geq b_{1}+b_{2}+\cdots+b_{k} .
\end{aligned}
$$
Find as many partitions of the number $n=20$ as possible such that none of the partitions dominates any other. Your score will be the number of partitions you find. If you make a mistake and one of your partitions does dominate another, your score is the largest $m$ such that the first $m$ partitions you list constitute a valid answer.