Define an operator acting on symmetric functions that has the family as eigenfunctions as
where n(µ) is the sum of µi(i-1).
These are some of the amazing conjectures associated with :
space of diagonal harmonics
If the coefficient of en(X)
is taken in this symmetric function, we have the definition of the q,t-Catalan
numbers as they are defined
in this web site.
If the coefficient of e1n(X) is taken in this symmetric function, we have (conjectured) the Hilbert series for the space of diagonal harmonics and the dimension of this space is conjectured to be (n+1)n-1, the number of rooted forests on n labeled verticies.
(a formula for the sign of this symmetric function is also conjectured)
It is not obvious from the definition that the operator nabla gives polynomial coefficients when it acts on the Schur symmetric functions. The proof of this theorem was the first proof that the q,t-Catalan numbers were polynomials in q and t. A resolution of either of the conjectures above would also imply that the q,t-Catalan numbers are polynomials with non-negative integer coefficients.
It seems that <H(k1(n-k))(X;t), h1n> is a q,t analog of the numbers (k+1)k-1 (n)(n-k)
For some more of the amazing properties of this operator see the references below: