I understand what the posets of partitions 4 and 5 are. I am having a hard time understanding dimensions though, as I do not fully understand what total orders and realizers are. I know that the dimension can not be larger than the width of the poset. The width of the poset of partition 4 is two and partition 5 is width 2.
I believe that the dimensions of the two posets is also 2, but I am having a hard time understanding why, and if the dimension is two, what are the two total orders that make this up.