Given a list, I want to find all possible ways to split it into sublists such that:
For example, for the input:
list = {1, 2, 3, 4, 5}; Some possible splits would be:
{{1, 2}, {2, 3}, {3, 4, 5}} {{1, 2}, {2, 3, 4, 5}} {{1, 2, 3}, {3, 4, 5}} ... {{1, 2, 3, 4, 5}} 
list = {1, 2, 3, 4, 5}; (Prepend @@@ Transpose[{#, Prepend[Most[#][[All, -1]], {}]}] /. {} -> Nothing) & /@ (TakeList[list, #] & /@ Select[Flatten[Permutations /@ IntegerPartitions[Length[list]], 1], #[[1]] != 1 &]) {{1,2,3,4,5}} {{1,2,3,4},{4,5}} {{1,2,3},{3,4,5}} {{1,2},{2,3,4,5}} {{1,2,3},{3,4},{4,5}} {{1,2},{2,3,4},{4,5}} {{1,2},{2,3},{3,4,5}} {{1,2},{2,3},{3,4},{4,5}} Or maybe simpler:
list = {1, 2, 3, 4, 5}; Split[#, Not@* SameQ] & /@ (Flatten[ list /. Thread[# -> Replace[#, x_ -> {x, x}, 1]]] & /@ Subsets[list[[2 ;; -2]]]) {{1,2,3,4,5}} {{1,2},{2,3,4,5}} {{1,2,3},{3,4,5}} {{1,2,3,4},{4,5}} {{1,2},{2,3},{3,4,5}} {{1,2},{2,3,4},{4,5}} {{1,2,3},{3,4},{4,5}} {{1,2},{2,3},{3,4},{4,5}} 
With[{n = Length[list] - 2}, FoldList[Prepend[#2, Last[#]] &, TakeList[list, Append[#, All]]] & /@ Differences /@ Join[ConstantArray[-1, {2^n, 1}], Subsets[Range[n]], 2]] {{{1, 2, 3, 4, 5}}, {{1, 2}, {2, 3, 4, 5}}, {{1, 2, 3}, {3, 4, 5}}, {{1, 2, 3, 4}, {4, 5}}, {{1, 2}, {2, 3}, {3, 4, 5}}, {{1, 2}, {2, 3, 4}, {4, 5}}, {{1, 2, 3}, {3, 4}, {4, 5}}, {{1, 2}, {2, 3}, {3, 4}, {4, 5}}}
Or
With[{n = Length[list]}, list[[# ;; #2]] & @@@ Partition[Join[{1}, #, {n}], 2, 1] & /@ Subsets[Range[2, n - 1]]] Another method:
list = {1, 2, 3, 4, 5}; Map[ Take[list, #] &, Partition[#, 2, 1] & /@ Flatten /@ Tuples[MapAt[Subsets, TakeList[list, {1, Length[list] - 2, 1}], 2]], {-2}] (* {{{1, 2, 3, 4, 5}}, {{1, 2}, {2, 3, 4, 5}}, {{1, 2, 3}, {3, 4, 5}}, {{1, 2, 3, 4}, {4, 5}}, {{1, 2}, {2, 3}, {3, 4, 5}}, {{1, 2}, {2, 3, 4}, {4, 5}}, {{1, 2, 3}, {3, 4}, {4, 5}}, {{1, 2}, {2, 3}, {3, 4}, {4, 5}}} *) A graph-based method:
list = Range[5]; With[{g = RelationGraph[Last@# == First@#2 & , Subsequences[list, {2, Length@list}]]}, Prepend[{list}]@ Flatten[MapApply[FindPath[g, #, #2, Infinity, All] & , Tuples[Map[Pick[VertexList[g], #, 0] &]@ Comap[{VertexInDegree, VertexOutDegree}]@g]], 1]] (*{{{1, 2, 3, 4, 5}} , {{1, 2}, {2, 3, 4}, {4, 5}} , {{1, 2}, {2, 3}, {3, 4}, {4, 5}} , {{1, 2}, {2, 3}, {3, 4, 5}} , {{1, 2}, {2, 3, 4, 5}} , {{1, 2, 3}, {3, 4}, {4, 5}} , {{1, 2, 3}, {3, 4, 5}} , {{1, 2, 3, 4}, {4, 5}}}*)
