Is [n] the set {1, ..., n}?

To form each X = {a_1, ..., a_k}, one is required to pick a certain number of elements of [n]: first, a_1, ..., a_k themselves and, second, c - 1 elements a_i + 1, ..., a_i + c - 1 for each 1 <= i < k. The remaining elements of [n] have to be spread between k + 1 intervals: from 1 to a_1, from a_k to n, and from a_i and a_{i+1}, 1 <= i < k. Ways to spread the remaining elements are in 1-1 correspondence with elements of X. You can use Stars and bars theorem.