$P=QRS$ decomposition of given list

Question

This is an extension of my previous post $P=QR$ decomposition of given list

Now for this time, I want to decompose arbitrary list $P$ into three pieces for ordered or non-ordered cases.

Based on the response of @Bill, I consider $P=QR'$ and make $R'=RS$ decomposition, but I realized my input $R$ is no longer a list but a array of lists.

For example, based on the code from @Bill,

P={a1,a2,a3,a4};
(*ordered*)
Q=Table[Take[P,i],{i,Length[P]-1}]
R=Table[Drop[P,i],{i,Length[P]-1}]
(*unordered*)
Q=Subsets[P,{1,Length[P]-1}]
R=Map[Complement[P,#]&,Q] 

To decompose $R$ into $R_1, R_2$, For each case of an array $Q$, I mean $Q[[i]]$, assign $R[[i]]$ as $P$ and then do the similar thing.

I want my code to do for the arbitrary given list, but this case, it seems not good.

Actually, I need $P=QRST$ decomposition but with $P=QRS$ I think I can handle $P=QRST$ decomposition easily.

---

For unordered case; following are my wrong trials using "for"

P = {a1, a2, a3};
Q = Subsets[P, {1, Length[P] - 1}]
R1 = Map[Complement[P, #] &, Q] 
For[i = 1, i <= Length[R1], i++,
R[[i]]=Subsets[R1[[i]], {1, Length[R1[[i]]] - 1}];
S[[i]]=Complement[R1[[i]], R[[i]] ]; ]

First of all, I notice my decomposition is totally wrong. Dividing the size of element of $Q$ is right, but the number is wrong. Even for three entries and dividing into three parts, the size two should be neglected but the above does not care about it. And Identifying R[[i]] is also totally wrong, because what Susbsets[R1[[1]],{1, Length[R1[[1]]] - 1}] = {{a2},{a3}}....

After @Bill 's response to this post, I realized to make it work for unordered partitions, there should be

 v=Map[TakeList[P,#]&,Flatten[Map[Permutations,IntegerPartitions[Length[P],{3}]],1]];

for the unordered subsets. First I think of splitting the given array into three; then realized the Mathematica splits the list in an ordered way. i.e., splitting {1,2,3} into three it only produce {{1},{2},{3}}

Answer 1

Check this very carefully to make certain it is correct for this example and any others.

P={a1,a2,a3,a4,a5,a6};
(*ordered*)
v=Map[TakeList[P,#]&,Flatten[Map[Permutations,IntegerPartitions[Length[P],{3}]],1]];
Q=Map[#[[1]]&,v]
R=Map[#[[2]]&,v]
S=Map[#[[3]]&,v]

which gives values of Q,R,S of

{{a1,a2,a3,a4},{a1},{a1},{a1,a2,a3},{a1,a2,a3},{a1,a2},{a1,a2},{a1},{a1},{a1,a2}}
{{a5},{a2,a3,a4,a5},{a2},{a4,a5},{a4},{a3,a4,a5},{a3},{a2,a3,a4},{a2,a3},{a3,a4}}
{{a6},{a6},{a3,a4,a5,a6},{a6},{a5,a6},{a6},{a4,a5,a6},{a5,a6},{a4,a5,a6},{a5,a6}}

There are lots of other ways of writing that, but I haven't found what I would believe is the simplest possible way of doing that.

Likewise for the unordered partitions.