Dr. A came to Dr. B asking for research ideas. Dr. B gave him a thesis, explained its value, why it's plausible, and how to test it. Dr. A liked this, got Student C to do the work, and managed the work.
Now, it's time to publish their results. How should they determine the lead authorship?
The lead author should be the person who carries out the bulk of the research work—including the planning, execution, and writing.
If the student C is the one responsible for carrying out the experiments and writing the paper, then C should be considered the primary agent of the paper.
Who should be the "final" author is a different matter. I would suspect that this really should be professor B, who had the original idea. The role of doctor A is only as research manager, so his role is nominally weaker than that of either B or C in this process. So, ultimately, barring other negotiations and arrangements to the contrary, it should be:
Student C, Doctor A, Professor B.
If there is some other distribution of roles beyond what is laid out above, it would need to be negotiated between everybody involved.
Note: In fields where lexicographic order is not used, the final author position also carries weight. It usually is the head of the research group or team that originated the project.
There is no easy answer. Providing an idea is certainly a good part of research but performing experiments, interpreting results and writing up the paper (discussion) are equally important (by equal I do not mean they are worth exactly the same, just important). There are several ways in which one can discuss the order based on evaluating the value of each component of the research process and then evaluate each persons contribution on each of these components. There have been a few posts (not necessarily in their entirety relevant to your question) here that discusses contributorship which I will not reiterate here. Please check the following (particularly links in the answers):
Paper contributions and first authorship
How should a student defend his 1st authorship in front of his advisor politely and effectively?
Authorship for paper based on my thesis
The point here is that none of the different aspects are alone enough for first authorship so one must weigh all components involved in the research process. It is of course harder to accomplish this "after the fact" but if all involved agree to try I am sure you can come up with a good author order.
Dr. A was a wise man, a negotiated such things when asking for the research ideas? :-)
Well, if B is more senior than A, then in most fields the order would be C, A, B. First authorship goes to the student, who did the most work. Last authorship goes to the most senior researcher. Then, the only position left for A is the second authorship. (Assuming for example that C is a grad student, A is a post-doc or something, and B is a professor).
heavy lifting being the most important of the two though (and in general the one doing the most work should get the first authorship).
In addition, there is the possibility of having a paragraph "Contributions". Some journals actually ask for such a statement.
If the discussion about the order of authors becomes difficult, this can calm down the discussion, also because writing down who did what often points out the order the authors should have.
The description is quite vague.
If I read the paper and I have questions about it, who should I ask?
That should be the first author and in most cases this will fit with the description of C.
The ideas of B are indeed important, but in most cases this is overlooked and underestimated. If B didn't publish his ideas or did anything about them, then A could claim he had the same ideas, and B only stated the evident (I have seen this many times). More often than not, the role of B is to appear in the acknowledgements of the paper, or not to appear at all, because the conversation between A and B may be considered either:
Basically B gave his advice as a gift to A, and now A has to decide whether this gift is to be returned or not. If it is returned then probably B will be the second author, if it is not returned, then B will not be an author. This basically depends on how A sees B, he may be an ally, an asset, a friend, a superior, a rival, an enemy, an annoyance...
I have discarded the option of B being the third author because I'm assuming A is more senior than B, if B is more senior than A then B should be the third and not the second author, this would also mean that B has more chances to appear as a coauthor. The reason is that the order from first to last author means the involvement in the work and the paper, but also on low-level work, while the last author is in the highest-level work, most abstract, theoretical, general, long-term-visioned, etc. Therefore the most senior is set to the last (even if someone else was less involved but involved enough to appear) to avoid suspicion on a less senior researcher (aka lesser being) giving high-level advice to a more senior researcher.
Most of the time the focus, the stress and the effort is not on how things are, but how do they look like. At least this is what I have seen up until now.
BTW: When I put examples I like to speak about Alex, Bernie, Cory, etc. I think this is easier to follow and I do also think that @order made a mistake between A and B. By your description, A seems to be more senior than B, the kind of senior researcher that manages human resources, that can pay C and that can tell C what to do. If that is the case, I doubt as well about the involvement of A in the paper and the work, while B and A can
