Imagine we are making necklaces with $n$ beads, each bead is a different color from all others. Let's say we make one necklace. If we make another necklace with the same $n$ differently colored beads, choosing each bead at random, what is the chance that exactly $k$ of the $n$ nearest neighbor pairs of beads from the original necklace will also be nearest neighbors in the second necklace?

Similarly, consider $n$ people standing in a queue. How many pairs of people standing next to each other in the first queue will also stand next to each other in a second queue, if the placements are reordered completely randomly? What is the form of $A(n,k)$, the number of ways to arrange the people such that exactly $k$ of the original $n$ are still neighbors?

I am interested in counting the number of these pairs as a function of $n$ in $S_{n}$ and certain subgroups. A generating function giving the fraction of the permutations having $i={0,1,2,..n}$ shared adjacent pairs would be particularly interesting to me, but terminology, references or general guidance might be more helpful so I could pursue this further on my own.