How to determine the probability of the repetition of a subsequence?

suppose to have a sequence s of length n in which appear k symbols. For example:

s=1,2,3,4,5,6,1,2,3,4,5,6,1,2,3,4,5,6,1,2,3,4,5,6

n=24

k= 6 (1,...,6)

Observe that each symbol appears exactly the same number of times. So from this sequence we could infer that is a uniformly at random distribution.

I'm looking for a way to determine the probability of a subsequence repetition. In particular, from the given example we'd have that s is composed by ss1,ss2,ss3,ss4, with each ss=1,2,3,4,5,6.

In the simplest example we may suppose that each symbol appears the same number of times as others, but I need a way to deal also with cases as the follow:

s1=1,2,3,4,4,1,2,3,4,4,1,2,3,4,4

in which the repetition is 1,2,3,4,4.

In general the problem may summarized as follows:

Given:

 - A sequence *s* of length *n* in which appear *k* symbols
 - An interval I=s_i,s_j belonging to s such that they exist at least two subsequence in the interval that are consecutive and are the same (1,2,3,1,2,3,1,2,3)

Determine the probability that the repetition is random or not.