Well, interesting discussion :). I didn't check the code as well, unfortunately, but I think I need to defend Steffen regarding the Opdyke paper. (Sry, this post is not on the matlab implementation as well)
First, I agree, this paper should not state results about weak compositions if it excludes them. Otherwise, it should tell the reader that these objects differ from what the paper is analysing. But, still, this appears as a minor flaw to me. It does not merit extended discussions, I feel :)
Much more severe is the issue about the "novel" combinatorial results the paper lists (as quoted by Steffen). Clearly, these results are elementary and can be found in any undergraduate textbook on combinatorics. One might argue that the author was just sloppy or is not an expert but he apparently takes these results directly from the papers he cites. This is clearly a form of plagiarism (unethical appropriation of another's work, to my understanding). This is a really severe issue, particularly in the US.