I am teaching a course in enumerative combinatorics this semester and one of my students asked for deeper clarification regarding the difference between a "combinatorial" and a "bijective" proof. Specifically, they pointed out that when one is proving the validity of a combinatorial identity by counting a set in two different ways, this is a different activity than giving an explicit bijection between two different sets. However, in combinatorics we often use the phrases "combinatorial proof" and "bijective proof" as synonyms, and I have often heard people use the phrase "bijective proof" regarding a "count in two different ways" proof of an identity.
It seems to me that often a combinatorial proof arises from describing one set in two different ways; implicit in a proof of the equality of the two descriptions of this set is the identity bijection from the set to itself. In this sense, one might regard all "combinatorial" proofs as "bijective," but I feel that I am on quite shaky ground with this. These thoughts have led me to the following questions:
Question 1: What are some examples of combinatorial situations where the same set can be described in two different ways but it is not at all clear that the two descriptions yield the same object?
Question 2: What are some examples of situations where two bijective proofs have been given for a theorem or identity where the bijections turned out to be the same, but proving their equivalence was non-trivial?
I would also appreciate opinions regarding the distinction, if any, between combinatorial arguments where one proves identities by describing a set in two different ways and combinatorial arguments where one sets up bijections between genuinely different sets of objects.
EDIT
Thanks for the answers and comments so far. Here are two examples that will hopefully clarify what I am asking. One example of a bijective proof between two different sets are showing that Dyck paths and nonnesting partitions are both Catalan-enumerated objects (even preserving the Narayana statistic with a good bijection). On the other hand, the identity $\sum_{k=0}^nk{n\choose k}=n2^{n-1}$ is usually proved by describing $k$-subsets of $n$ with a distinguished element in two different ways: in the first way, pick the set then specify the element; in the second way, pick the element then specify the rest of the set. These are both referred to as bijective or combinatorial proofs, yet somehow they each have a different feel to them. In the second case, it is pretty easy to see that the two descriptions of these objects yield the same set of objects, but surely there must be more situations where the same set is described in two different ways and the equivalence of their descriptions is difficult to ascertain. Similarly, there must be times where there are several bijections between different sets, like the first example, where the bijections are the same but not obviously so. What I am wondering about are examples of these two situations.
A non-combinatorial example of an answer to Q1 is the compact-group vs reflection group definition of the Weyl group of a semi-simple Lie Algebra, where it isn't immediately clear that the same group is obtained. However, I am looking for more combinatorial examples.
Best Answer
Here is a fact that I find interesting. Strictly speaking it is to say that a certain number counts three completely unrelated things, but it seems easy to make these into a combinatorial problem or to define an explicit bijection between two (or three) sets.
So, the concrete example is $168$, which
(1) the number of hours in a week;
(2) the number of primes under $1000$; and
(3) the size of the smallest simple group of Lie type.
$168$ is also $4 \times 42$, where $42$ is that famous number Douglas Adams wrote about. (How did he know?) Which is of course the reciprocal of the smallest positive number that can be written as $$1-\frac 1a -\frac 1b -\frac 1c$$ with $a,b,c\in \mathbb N$ and coincidentally (and relatedly) the largest size of the automorphism group of $C$ a smooth projective curve of genus at least $2$ is $42\cdot {\rm deg} K_C$ and of $S$ a smooth projective surface of general type is $(42 K_S)^2$.
So I guess one could add that $168$ is also
(4) the maximum value of $\frac 4{1 -\frac 1a -\frac 1b -\frac 1c}$ with $a,b,c\in \mathbb N$; and
(5) the largest possible size of the automorphism group of a smooth complex projective curve of genus $3$,
but I admit the last two are a little artificial...