We have a complete graph $K_n$ with $n$ vertices.
Then the number of distinct 1-factorizations of a $K_n$ graph for $K_2$, $K_4$, $K_6$, $K_8$ is 1, 1, 6, 6240 respectively. (see A000438).
I wanted to understand these numbers, but I failed already for the example $K_4$. In my attempts, I would get 3 different 1-factorizations, not one:
Could you please tell me what I misunderstand?
A 1-factor is a spanning subgraph, while a 1-factorization of $K_n$ is the partition of $K_n$ into multiple 1-factors.
In the example given in the question, $K_4$ is partitioned into three 1-factors, but there is only one unique way to do that.
As another example, there are 6 ways to 1-factorize $K_6$ into 5 1-factors, as illustrated in the figures below.