@acer 

Wow, this is exactly what I needed. Really appreciate you taking the time to help!

@acer 

Thanks. But what if I want the procedure to accept both Groupings(4, 2) and Groupings([1, 2, 3, 4], 2) as valid input?

Thanks to @dharr for the quick and helpful response, it solved my problem perfectly. Also, thanks to @acer for the additional suggestion and simplification. Your answers were very helpful!

I'd like to add an additional requirement to the original problem::

Example:

Groupings(4, 2)  or  Groupings([1,2,3,4], 2)

Expected output:

[[[[1, 2], 3], 4], [1, [[2, 3], 4]], [[1, [2, 3]], 4], [1, [2, [3, 4]]], [[1, 2], [3, 4]]]

Would you mind suggesting how I could modify your solutions?

(18) Missing solutions in isolve

It appears to be a bug.

The 2 solutions represent two phases of motion under gravity, and both should be returned (as in Maple 7), irrespective of the sign of v__0.

Here is the output from Maple 7:

(17a) Strange Plot Behavior (Problem Solved)

(17) Strange Plot Behavior

(12a) Expressions (1) and (2) hold only when n is an integer, regardless of whether p>1 or p<1 , making the assumption "n::integer" superfluous.

(16) Round brackets missing in the Layout palette.

@acer 

In example (12), Expressions (1) and (2) hold only when n is an integer, regardless of whether p>1 or p<1 . 

In example (13), n only appears in the upper limits of the sigma notation. When n is not an integer, would it be appropriate to treat n as floor(n), so as to ensure that the upper limits are integers?

(15) Expression (5) should simplify to 1

(14) Both expressions should simplify to 1.

(13) The expression should simplify to 1.

(12) Is the assumption really necessary? Mathematica gives the result in one step without that assumption.