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 [17] 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 [1] 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 [4] 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 [2]: 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.