1978: I took an excellent ugrad course in combinatorics from James C Frauenthal (he sometimes wrote his name as the biniomial cofficient (J choose F)) and he covered Ferrer's diagrams. They are a nice way to prove equalities about types of partitions. See here for a definition and a few examples. I have this (possibly false) memory that there were LOTS of partition theorems proven nicely with Ferrer's diagrams.
Fast forward to 2021:
2021: My TA Emily needs a topic to cover in Honors Discrete Math. I have this memory that there were LOTS of theorems about partitions proven with Ferrer's diagrams. We look at many websites on Ferrer diagrams and find only TWO examples:
The numb of partitions of n into k parts is the numb of partitions of n into parts the largest of which is k.
The numb of partitions of n into \le k parts is the numb of partitions of n into parts the largest of which is \le k
We DO find many theorems about partitions such as this corollary to the Rogers-Ramanujan theorem:
The numb of partitions of n such that adjacent parts differ by at least 2 is the numb of partitions of n such that each partition is either \equiv 1 mod 5 or \equiv 4 mod 5.
This is a HARD theorem and there is no Ferrer-diagram or other elementary proof.
SO, I have one MEMORY but the reality seems different. Possibilities:
1) My memory is wrong. There really are only 2 examples (or some very small number).
2) There are other examples but I can't find them on the web. I HOPE this is true--- if someone knows of other ways to use Ferrer diagrams to get partition results, please comment.