Finding the Cardinality of Set C: A Problem in Subsequence Coverage

[grigor]

I have faced the following problem recently:

We have a sequence A of M consecutive integers, beginning at A[1] = 1: 1,2,...M (example: M = 8 , A = 1,2,3,4,5,6,7,8 )

We have the set T consisting of all possible subsequences made from L_T consecutive terms of A, which do not overlap. (example L_T = 3 , subsequences are {1,2,3},{4,5,6},{7,8,9},...). Let's call the elements of T "tiles".

We have the set S consisting of all possible subsequences of A that have length L_S. ( example L_S = 4, subsequences like {1,2,3,4} , {1,3,7,8} ,...{4,5,7,8} ).

We say that an element s of S can be "covered" by K "tiles" of T if there exist K tiles in T such that the union of their sets of terms contains the terms of s as a subset. For example, subsequence {1,2,3} is possible to cover with 2 tiles of length 2 ({1,2} and {3,4}), while subsequnce {1,3,5} is not possible to "cover" with 2 "tiles" of length 2, but is possible to cover with 2 "tiles" of length 3 ({1,2,3} and {4,5,6}).

Let C be the subset of elements of S that can be covered by K tiles of T.

Find the cardinality of C given M, L_T, L_S, K.

Any ideas would be appreciated how to tackle this problem.

 

[NateTG]

It may make sense to look at the union of each subset of K tiles, and then take power sets of their union.