TITLE: What is an explicit bijection in combinatorics?
QUESTION [44 upvotes]: A standard way of demonstrating that two collections of combinatorial objects have the same cardinality is to exhibit a bijection between them. Browsing through some examples (here, there, yonder) quickly reveals that combinatorialists call such bijections explicit, presumably to differentiate them from other less palpable kinds of bijections. Wikipedia speaks of the method of bijective proof.
It seems that we have here a typical example of an informal mathematical notion that is quite familiar to most mathematicians, however it is difficult to pin down a proper and satisfying mathematical definition. I asked the local combinatorialists and did not really get a good answer.
Question: What is a proper mathematical definition of an explicit bijection?
Often we ask for an explicit bijection between two families of combinatorial objects, i.e., bijections $b_n : A_n \to B_n$, one for each $n \in \mathbb{N}$. Here $(A_n)_n$ and $(B_n)_n$ are two families of combinatorial objects, parametrized by $n$. The parameter need not be a single number.
Here are some unsatisfactory answers:

"A bijection is explicit if it is computable." This definition is too wide, because it allows silly algorithms that order combinatorial objects according to the layout of the sequences of bits that represent them, and use the order to establish a bijection. Bit representations typically have nothing to do with the combinatorial content of the objects under consideration.
"A bijection is explicit if it can be written down as an expression." This takes us back several centuries in terms of level of mathematical abstraction, and also varies a lot depending on what expressions are allowed. We really should be looking for a combinatorially meaningful notion, not a syntactic surrogate.
"A bijection is explicit if we can give a constructive proof of its existence." Well, a constructive proof certainly guarantees that a computable bijection exists, and can moreover be extracted from the proof, but this still feels too permissive. For example, we can always compose an explicit bijection so obtained with a computable automorphism of one of the sets, and still have a constructive proof. But such an automorphism could completely obfuscate the combinatorial structure of the set.
"Well-order $V_\omega$ (as all combinatorial objects easily live in it) and take the first bijection under the well ordering." Only a set theorist would have such thoughts. Again, we should strive for a definition which will be accepted as natural by combinatorialists.

Let me also say that I would prefer to not generalize the question to "what is an explicit thing?" At least in combinatorics "explicit bijections" are a well-established and useful notion, whereas mathematicians in general do not posses a universally agreed upon notion of "explicit thing".
Supplemental: After having a look at Igor Pak's paper, I am somewhat convinced that computational complexity plays a certain role, but it cannot be the only answer (as Pak himself notes). For example, an explicit bijection may require factoring of numbers, which I feel most people would find unproblematic even though the computational complexity of factoring is not resolved.

REPLY [6 votes]: I would like to adopt a slightly contrarian viewpoint:

There is no formal mathematical definition of "explicit bijection."

Of course, I can't formally prove this assertion, but I would say that the reason you're having trouble finding a satisfactory formal definition is precisely because there isn't one.
A similar issue comes up in the Razborov–Rudich theory of natural proofs.  Quoting from their paper:

Note that the definition of a natural proof, unlike that of a natural combinatorial property, is not precise.  This is because while the notion of a property being explicitly defined in a journal paper is perfectly clear to the working mathematician, it is a bit slippery to formalize.  This lack of precision will not affect the precision of our general statements about natural proofs because they will appear only in the form "there exists (no) natural proof…", and should be understood as equivalent to "there exists (no) natural combinatorial property $C_n$…"

Taking a cue from the above, I think that what may be more productive than trying to pin down an exact definition of an explicit bijection is finding sufficient conditions for being an explicit bijection.  I say this because I have a secret agenda: performing automated searches for explicit bijections.
For this purpose, I think it would be useful to compile a list of "atomic" components of an explicit bijection and say that if one combines no more than $x$ such components in certain specified ways, then the resulting bijection (or map, if we don't know in advance that it is bijective) is explicit.
By the way, here's an analogous issue from recreational mathematics.  What does it mean to say that a Sudoku puzzle (let's assume that it has a unique solution) can be "solved without guessing"?  I don't think that there is a canonical answer to this question, because what looks like guessing to you or me might just be a "standard trick" to a sufficiently powerful brain.  On the other hand, it is possible to compile a specific long list of known tricks, and then you can automate the generation of "Sudokus solvable without guessing."