Assume a sequence $S$ of numbers out of the set $N={1..n}$.
Example: $S = "123312"$
Set of all pairs would be: $M = (2,3),(3,3), (3,1)$
Not in $M$:
$(1,1)$ : not occuring in the sequence next to each other.
$(1,2)$ : occuring twice
Numbers are order-dependent. $(1,2)$ != $(2,1)$
What is the probability that all pairs $(a$,$b)$ out of the set $M$ that occur only once in $S$ are evenly distributed over the length of the sequence $S$? How do i model "evenly distributed" over $S$ best?
(In reality i would like an even distance between the pairs $(a$,$b)$. Or at least the probability for that.) Should i model it as part of Quintiles with bins in which the pairs fall(like i highlighted in this example? What would be the mathematical term for such a distribution?
The arrow represents the sequence $S$ made up of instances of $N$ of length $L$. The yellow circles denote the position $p$ of a pair $(a$,$b)$ that only once occurs in $S$. The horizontal arrow denotes the distance $D$ between two yellow circles.
I also would like to plug into this different probabilities from the set N={1..n} with pN=p(1..n).
What i have tried so far is computing the probability that a pair occurs in a certain part of the sequence and not in the other parts. I them sum this up over all possible pairs. However the equations get pretty nasty somehow and mathematica cannot simplify them by much. Pretty sure, that there should be a solution that is not too complicated.