So it would be the expected wait time of that occourance

Are you sure it is the same as the longest expected occourance of H coin flips? It is allowed for the common substring to be HTHHTHT.. in that analogy

Yes! @PatDevlin.. Assuming q = 2 and N = 100 (for example) what would be the expected longest common substring

Maybe I should remove the increasing part. What I've meant with increasing is that as N is increasing, the expected average mean length of the longest common substring grows as what function f(N, q) ?

I've added additional examples to clarify. Also added "another" way to state the given problem. Is everything clear now?

Please state if anything else is left unclear and needs clarification so I can edit my post!

Sorry all the misunderstandings are due to my English.. I've edited and corrected everything. Permutation was a bad word to use. With concrete I mean any paper which I can use for referencing (best'd be published) i.e. "this is proven in ... [1]" Refferences: [1] this and this paper, pp. xyz.. @MattF.

thw longest consecutive piece in both strings is ABC