Suppose you have a set of $N$ random integers. Each integer is limited to the range from $0$ to $99$.
You then consider each subsequent neighbor as "ascending" or "descending", counting the number of consecutive neighbors who are also of the same kind. ( i.e. $1234$ is a "run" of $3$ ascending neighbors; $123431$ is a ascending run of $3$ followed by a descending run of $2$ ). As an assumption, the problem considers both ascending and descending to be functionally symmetrical/same. We can only consider runs of one direction.
In the set of $N$ members, what is the probability of any Run of length $L$?
i.e. In the set of $N$, how many runs of length $1$ should there be? Is there a formula for Runs-of-Length given $N$ and the range of values of the integers?
I feel that there should be a "simple" formula/algorithm, but I cannot figure it out and I have not found any similar information with searches.
An EXAMPLE data set, a specific series generated the following "rising" runs.
$N$ = $67108864$
[Length]= num of occurs
[0]= 0
[1]= 13926757
[2]= 6144339
[3]= 1779212
[4]= 391946
[5]= 71071
[6]= 10716
[7]= 1390
[8]= 164
[9]= 16
[10]= 3
[11]= 0
[12]= 0
Total number of Runs = $22325614$
As you can see, the distribution appears to be some sort of bell perhaps. However, I am unable to figure out what kind and what the PDF is. All series I generate from various "random" sources seem to yield similar results, but not exact ( of course ).
Thanks in advance!