Is there a name for this set/array operation?

Question

Given the input array

[a,b,c,d,e]

and a 'join' function (a,b) => (a+b)

my code returns the following array of arrays, containing each possible variation obtained by applying the join function to various pairs of elements whilst maintaining the order:

[
  [a,b,c,d,e],
  [a,b+c,d,e],
  [a,b+c+d,e],
  [a,b,c+d,e],
  [a+b,c,d,e],
  [a+b,c+d,e],
  [a+b+c,d,e],
  [a+b+c+d,e],
  [a,b,c,d+e],
  [a,b+c,d+e],
  [a,b+c+d+e],
  [a,b,c+d+e],
  [a+b,c,d+e],
  [a+b,c+d+e],
  [a+b+c,d+e],
  [a+b+c+d+e],
]

Visually, what I'm trying to do is this:

The code works but I have no idea what to call it - and would like to use a name that other developers familiar with this operation will understand, should such a name exist. It's not a power set but it is something similar... does this particular set/array operation have a name?

EDIT: OK. They're not permutations; permutations would all be 5-element arrays in different orders [[a,b,c,d,e], [e,d,c,b,a], [a,d,b,c,e], ...]

They're not partitions, since any subset can only contain adjacent elements of the input. - in other words, partitions would allow this:

(This probably arises from pure set theory having no notion of an ordered set.)

They're not combinations since every element of the output uses every member of the input set exactly once.

I think myArray.OrderedPartitions((a,b) => (a+b)) is probably a suitably succinct and explanatory.

Answer 1

As mbeckish said in a comment, those sets are (once an order on the original set is fixed) isomorphic to order-dependent integer partitions, which apparently are commonly referred to as compositions. There are exactly 2^n-1 compositions of every set. For every 1 ≤ k ≤ n, there are exactly (n-1) choose (k-1) compositions of n elements into k sets, preserving the order of the set you started out with. To visualize it, think of the elements of your set put in order and a space between elements that are neighbours in that order; think of your example as A|B|C|D|E. You will notice that there are exactly n-1 possible borders. To create a k-composition, you need only choose k-1 of those possible borders, which may or may not be the way you generated your sets. Summing all (n-1) choose (k-1) for k from 1 to n then gives us 2^n-1 as the number of possible compositions.