Trying to derive a formula to quantify the degree to which objects in an original order are scrambled upon some amount of repeated random handling.
Say there are $n$ spheres labeled with labels $1$ thru $n$ each of radius $r$ sitting in a straight line ordered left to right 1 thru $n$ on a table top. Person $A$ starting from the leftmost sphere places them into a cylindrical container of radius $R$ where $R>r$, one by one, piling up as they go, starting with leftmost sphere $1$ and ending with the rightmost sphere $n$ placed in last.
Person $B$ then randomly takes a topmost sphere from the cylinder and places it on the table all the way to the right and continues to take the topmost sphere and place it to the left of the previous pick until they are all on the table.
At this point, the spheres on the table are in a similar but slightly different order than they were before person A placed them in the cylinder.
Question is, how would one quantify the degree to which the order is scrambled after a series of $N$ such random place and random pick pairs from the same cylinder have taken place?
Then same question, but with a new cylinder of uniformly random radius ranging from $R=3r$ to $R=(n/4)r$, that is from uniform distribution $U[3r,(n/4)r]$, for each round of placing and picking.