KAIST Discrete Math Seminar

In this talk, we consider the combinatorial formula for the Schur coefficients of the integral form of the Macdonald polynomials. As an attempt to prove Haglund’s conjecture that ⟨J_λ[X;q,q^k]/(1-q)ⁿ,s_μ(X)⟩∈ℕ[q], we have found explicit combinatorial formulas for the Schur coefficients in one row case, two column case and certain hook shape cases. Egge-Loehr-Warrington constructed a combinatorial way of getting Schur expansion of symmetric functions when the expansion of the function in terms of Gessel’s fundamental quasi symmetric functions is given. We apply this method to the combinatorial formula for the integral form Macdonlad polynomials of Haglund-Haiman-Loehr in quasi symmetric functions to get the Schur coefficients and prove the Haglund’s conjecture in more general cases.

The cycle counting rook numbers, hit numbers, and q-rook numberes and q-hit numbers have been studied by many people, and Briggs and Remmel introduced the theory of p-rook and p-hit numbers which is a rook theory model of the weath product of the cyclic group C_p and the symmetric group S_n.
We extend the cycle-counting q-rook numberes and q-hit numbers to the Briggs-Remmel model. In such a settinig, we define multivariable version of the cycle-counting q-rook numbers and cycle-counting q-hit numbers where we keep track of cycles of pernutation and partial permutation of C_p wearth product with S_n according to the signs of the cycles.
This work is a joint work with Jim Haglund at University of Pennsylvania and Jeff Remmel at UCSD.