44. 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.
Comments: Computer searches have found that the maximum possible number of partitions is 20 , achieved e.g. by the following:
$$
\begin{array}{c}
9+1+1+1+1+1+1+1+1+1+1+1 \\
8+2+2+1+1+1+1+1+1+1+1 \\
7+4+1+1+1+1+1+1+1+1+1 \\
7+3+3+1+1+1+1+1+1+1 \\
7+3+2+2+2+1+1+1+1 \\
7+2+2+2+2+2+2+1 \\
6+5+2+1+1+1+1+1+1+1 \\
6+4+3+2+1+1+1+1+1 \\
6+4+2+2+2+2+1+1 \\
6+3+3+3+2+1+1+1
\end{array}
$$
$$
\begin{array}{c}
6+3+3+2+2+2+2 \\
5+5+4+1+1+1+1+1+1 \\
5+5+3+2+2+1+1+1 \\
5+5+2+2+2+2+2 \\
5+4+4+3+1+1+1+1 \\
5+4+4+2+2+2+1 \\
5+4+3+3+3+1+1 \\
5+3+3+3+3+3 \\
4+4+4+4+2+1+1 \\
4+4+4+3+3+2
\end{array}
$$
"Chain Lengths in the Dominance Lattice," by Edward Early, 2002 the maximum number of mutually nondominating partitions of $n$ is roughly on the order of $e^{\pi \sqrt{2 n / 3}}$ (in fact, the ratio of the total number of partitions to $n$ to the maximum number of nondominating partitions has polynomial order of magnitude), although a proof by explicit construction is not known.