Hi, thanks Roloph, for supporting my view. Yes, I also think the formulas might be plagiarism but, after all, this whole paper is sometimes quite inaccurate and I'm sure the authors did not understand the difference between weak compositions and compositions (otherwise, why state results on weak compositions?) or even might not have noticed that their formulas are explicitly stated in referenced work (incredible as it seems) ...
I'm sorry, but I need to make another statement, to again illustrate the deficiency of the Opdyke paper. On p. 13, it verbally says:
"Although the recursive nature of (1)-(4) makes these formulae less convenient than, say, a simple combinatoric equation or sum, they still provide closed form solutions to problems which had none before [...]"
Now, equation (1) is just the Fibonacci identity for integer compositions, which is well-known since ages. Kimberling has it; every elementary text-book has it. Eq. (2) is nothing. Eq. (3) is just the Vandermonde identity for restricted integer compositions; this is well-known for ages as well (besides being trivial): The Heubach and Mansour (2004) you quote paper has it on p. 2 without even giving a proof. And equation (4) is just the summation of (3) --- this is entirely trivial.
And you want to sell these results as new? And the journal editor and the referee did accept this? And you find my comment regarding the quality of that journal subjective? Excuse me, sir, something has gone wrong with parts of your publication ...
And your Kimberling citation: You cannot simply reference "a related result" using the same notation as you use. This is just bad style.
Finally, regarding your notation and your abuse of "explicitness": Compare your cumbersome notation with the elegant notation in Heubach and Mansour (2004) to see the differences. You would just have needed to state in a single place (usually called "Notation" or so): Henceforth, let always 1 \le a \le b \le n. [All other cases are trivial anyway]
1) This implementation is false. Period. It is. There are verifiable bugs in the code. They may not matter much and maybe they don't matter at all if a is greater than zero. But this implementation is not the same as the pseudocode given in that very precise and exact Opdyke paper.
This has only referred to the implementation (but J.D. has apparently something to say on this well although he never checked the implementation, as he says).
And what I'm saying now additionally is (well, I always had that opinion): The Opdyke paper is confused and confusing, particularly in its notation (we're through this ...) and also in some statements; e.g., where it says there is an open problem of finding a closed-form solution to the restricted integer composition problem and where it claims that it provides such a solution (where is this solution? And don't say it's formulas (1),(2),(3), and (4) --- they've been around for ages ...)
@J.D.: Discussions are very personal now. I just want to add the following.
(1) Your paper is not clear and it is not consistent. You apparently explicitly base your paper on Kimberling, refering to his notation and to his results in several instances. Throughout, you use the notation c(n,k,a,b) to refer to integer compositions of n with k parts, each between a and b. A very superficial check reveals that you use the notation c(n,k,0,b) at least twice, primarily, when referring to to theoretical results. You should not mention weak compositions, as you do, if you feel that such compositions are "so rarely used not to merit a mention". And you should also not mention them if you want to explicitly exclude them ...
You see, here is the sentence: "Following Kimberling's (2001) notation, if we restrict the values of those compositions by a minimum value of a and a maximum value of b for c(n,k,a,b), then with b=infinity [...] we have c(n,k,a,infinity)=c(n-ka,0,infinity)"
That settles the point, I feel ...
(2) I know research groups that like weak compositions. But maybe they do not merit a mention, as you say.
(3) You should be happy to hear that your algorithm works for a=0 as well. The reason is, it seems to me, that you base your work on Kimberling and he seems to like the a=0 case as well ;)
(4) If your algorithm works even if some of its aspects are not considered (as it is claimed to hold true in this implementation), then maybe you should recheck if your algorithm cannot be simplified or generalized.
(5) To me, it seems that you mix up "objectivity" with "applause".
@Oscar: Yes, some people make a distinction between weak compositions and compositions ...
But, the paper you cite, the Updyke algorithm, deals with restricted integer compositions, which usually restrict parts to lie within an arbitrary subset of the nonnegative integers or within an interval. For instance, read the background section of the cited Updyke paper. Or read the introduction of Kimberling, which is cited by Updyke: http://www.fq.math.ca/Scanned/39-5/kimberling.pdf.
(In particular, note how Updyke is unclear on which definition to use: the Introduction says something different than the background section and than Kimberling)
What's more problematic is that the Updyke algorithm works for any a>=0, but this implementation does not, because there are a few minor programming mistakes (but with major consequences): I listed them; in particular, in two lines a condition that has the form
if A \le B
in the Updyke paper reads as
in the implementation, which is clearly wrong.
Defining compositions for nonnegative integers is standard, allowing negative numbers is not.