The only known combinatorial proof
of of the unimodality of q-binomial coefficients is given by O'Hara in [O'H] (see also [SZ, Zei]).
Quinn: Proofs that Really Count, The Art of Combinatorial Proof
, Mathematical Association of America, Washington, D.
Bressoud  gave a combinatorial proof
of Schur's 1926 theorem by establishing a one-to-one correspondence between the two types of partitions counted in the theorem.
In math, such a tangible breakdown is called a combinatorial proof
Thanks to it, we obtain a combinatorial proof
of what was left as an open question in : the symmetric distribution of the initial rise and lower contacts of intervals.
for his project "A Combinatorial Proof
of Seymour's Conjecture for Regular Oriented Graphs with Almost Regular Outsets O'a and O"a.
Such a map would provide a combinatorial proof
of the major index side of (2).
In Section 6 we propose some open problems which will lead to a combinatorial proof
of the Selberg integral formula.
They comment that the role played by hyperbolic geometry in this problem, whose statement is purely combinatorial, may seem mysterious and they ask for a combinatorial proof
of their existence result.
In this section we prove the above theorem combinatorially, thus providing the first combinatorial proof
of their result.
Note that one can also give a direct combinatorial proof
similarly as in .
In Section 4 we apply the theorem to give a combinatorial proof
of an identity satisfied by [alpha](n; [k.