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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 next necklace?

Similarly, what about $n$ people sitting at a circular table in the morning and in the evening. How many pairs of people sitting next to each other in the morning will also sit next to each other in the evening, 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.

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.

Imagine we are making bracelets with $n$ beads, each bead is a different color from all others. Let's say we make one bracelet. If we make another bracelet 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 bracelet will also be nearest neighbors in the next bracelet?

Similarly, what about $n$ people sitting at a circular table in the morning and in the evening. How many pairs of people sitting next to each other in the morning will also sit next to each other in the evening, 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.

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 next necklace?

Similarly, what about $n$ people sitting at a circular table in the morning and in the evening. How many pairs of people sitting next to each other in the morning will also sit next to each other in the evening, 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.

Imagine we are making bracelets with $n$ beads, each bead is a different color from all others. Let's say we make one bracelet. If we make another bracelet 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 bracelet will also be nearest neighbors in the next bracelet?

Similarly, what about $n$ people sitting at a circular table in the morning and in the evening. How many pairs of people sitting next to each other in the morning will also sit next to each other in the evening, 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.

Imagine we are making bracelets with $n$ beads, each bead is a different color from all others. Let's say we make one bracelet. If we make another bracelet 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 bracelet will also be nearest neighbors in the next bracelet?

Similarly, what about $n$ people sitting at a circular table in the morning and in the evening. How many pairs of people sitting next to each other in the morning will also sit next to each other in the evening, 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.