This framework will be illustrated by a simple combinatorial proof
and interpretation of Taylor's formula in Section 2.1 and equations (5) and (6) in Sections 2.2 and 3.
A combinatorial proof
of Theorem 1.2 has been also obtained in  for the non-factorial case, as well as an analogous determinant formula for skew flagged Grothendieck polynomials, special cases of which arise as the Grothendieck polynomials associated to 321-avoiding permutations  and vexillary permutations.
It would be nice to have a direct combinatorial proof
of Corollary 4.
Quinn: Proofs that Really Count, The Art of Combinatorial Proof
, Mathematical Association of America, Washington, D.C., 2003.
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]).
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
. Ramanujan's work, and Ono's after it, relied on more-abstract proofs of divisibility.
Zeilberger, A combinatorial proof
of Bass's evaluations of the Ihara-Selberg zeta function for graphs, Trans.
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.
We can give a similar combinatorial proof
of Theorem 4.1 in the special case when t = 2.
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.