I am interested in computing the exact number of comparisons that are needed to sort a list. See this wikipedia article.
Up to $n=15$, we know how many comparisons between elements one must make to be sure the list is sorted. This sequence of numbers can be found here. The entry in OEIS does not mention exactly which elements should be compared (and when), though. I guess this is mentioned in the article it refers to, but I am not certain, since I don't have the moment and they don't seem to be available for free.
But let's suppose we do know which elements should be compared. Then we can write up a list of pairs of these elements. Then, we could visualize these pairs by drawing a graph, in which each node represents an element of the list (that ought to be compared with another element in order to sort the list) and each vertex between two elements represents a comparison between two elements of the list that must be compared in order to sort the list.
Q: I am wondering if we could deduce anything from these graphs, by putting them side by side. Do any patterns emerge? Could we extrapolate how the graphs for $n>15$ might look like?