James Haglund

A combinatorial formula for the character of the diagonal coinvariants

Author(s): Haglund, J; Haiman, M; Loehr, N; Remmel, J B; Ulyanov, A | Abstract: Let Rn be the ring of coinvariants for the diagonal action of the symmetric group Sn. It is known that the character of Rn as a doubly graded S-module can be expressed using the Frobenius characteristic map as nabla en, where en is the n-th elementary symmetric function and nabla is an operator from the theory of Macdonald polynomials. We conjecture a combinatorial formula for nabla en and prove that it has many desirable properties that support our conjecture. In particular, we prove that our formula is a symmetric function (which is not obvious) and that it is Schur positive. These results make use of the theory of ribbon tableau generating functions of Lascoux, Leclerc, and Thibon. We also show that a variety of earlier conjectures and theorems on nabla en are special cases of our conjecture. Finally, we extend our conjectures on nabla en and several on the results supporting them to higher powers nablam en.

A combinatorial formula for Macdonald polynomials

Abstract: We prove a combinatorial formula for the Macdonald polynomial

Conjectured statistics for the q,t-Catalan numbers

\tilde{H}_{\mu }(x;q,t)

A conjectured combinatorial formula for the Hilbert series for diagonal harmonics

which had been conjectured by Haglund. Corollaries to our main theorem include the expansion of

A combinatorial model for the Macdonald polynomials

\tilde{H}_{\mu }(x;q,t)

A positivity result in the theory of Macdonald polynomials

in terms of LLT polynomials, a new proof of the charge formula of Lascoux and Schutzenberger for Hall-Littlewood polynomials, a new proof of Knop and Sahis combinatorial formula for Jack polynomials as well as a lifting of their formula to integral form Macdonald polynomials, and a new combinatorial rule for the Kostka-Macdonald coefficients

Quasisymmetric Schur functions

\tilde{K}_{\lambda \mu }(q,t)

Stable multivariate Eulerian polynomials and generalized Stirling permutations

in the case that

Generalized Rook Polynomials

\mu

Theorems and Conjectures Involving Rook Polynomials with Only Real Zeros

is a partition with parts