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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?

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P[n] will return the set you are asking

P[n_] := Partition[Join[{0}, #, {n}], 2, 1] & /@ Subsets[Range[n - 1]]

P[4]

{{{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}}}

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  • $\begingroup$ Thanks, it works well!!! $\endgroup$
    – Mark_Phys
    Apr 1, 2019 at 9:08
  • $\begingroup$ @user10709 I'm glad I helped! $\endgroup$
    – ZaMoC
    Apr 1, 2019 at 9:10
  • $\begingroup$ Of course, you can combine the functions so that only one application of Map[] is needed: Partition[Join[{0}, #, {n}], 2, 1] & /@ Subsets[Range[n - 1]]. $\endgroup$
    – J. M.'s slightly less busy ♦
    Apr 1, 2019 at 14:35
  • $\begingroup$ @J.M.isslightlypensive yes, you are right $\endgroup$
    – ZaMoC
    Apr 1, 2019 at 14:52

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