First I give an example. For an integer set $(0,1,2,3,4)$, there are eight kinds of subdivision or partition like this $$(0,4);\\~~(0,1)(1,4);~~(0,2)(2,4);~~(0,3)(3,4);\\ (0,1)(1,2)(2,4);~~(0,1)(1,3)(3,4);~~(0,2)(2,3)(3,4);~~\\(0,1)(1,2)(2,3)(3,4); $$
For a more general set $(0,1,2,...,n)$, there are $2^{n-1}$ kinds of partition.I believe that there must be a special name for this kind of partition mathematically. How can I realize it in MMA?