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Define an operator acting on symmetric functions that has the family as
eigenfunctions as

= t^{n(µ)}q^{n(µ')} |

where n(µ) is the sum of µ_{i}(i-1).

These are some of the amazing conjectures associated with :

Conjecture 1

_{n}(X)
is the Frobenius series for the
space of diagonal harmonics |

If the coefficient of e_{n}(*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 e_{1}n(*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.

Conjecture 2

_{µ}(X)
is totally Schur positive or totally Schur negative
(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.

Conjecture 3

_{µ}(X)
is totally Schur positive or totally Schur negative |

Conjecture 4

_{µ}(X;1/t)
is totally Schur positive or totally Schur negative |

Conjecture 5

_{µ}(X;t)
is Schur positive |

It seems that <H_{(k1}(n-k)_{)}(X;t),
h_{1}^{n}> 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:

- Explicit Plethystic Formulas for the Macdonald q,t-Kostka Coefficients (A. Garsia, M. Haiman and G. Tesler)
- Identities and Positivity Conjectures for Some Remarkable Operators in the Theory of Symmetric Functions
- A Remarkable q,t-Catalan Sequence and q-Lagrange inversion, J. of Alg. Comb. V. 5 (1996) pp. 191-244.

( F. Bergeron, A. M. Garsia, M. Haiman and G. Tesler ), submitted, 60 page

(A. Garsia and M. Haiman)