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Maple program for computing the q,t-Catalan numbers

The (q,t)-Catalan sequence is defined by setting

where the sum is over all partitions of *n*. All products and sums
in the µ^{th} summand are over the cells
of µ and the parameters
*l*,
*l'*,
*a*,
*a'*
denote the *leg*, *coleg*,
*arm* and
*coarm* of agiven
cell. That is, for a given cell *s*, when µ
is depicted bythe French convention (see the figure below) *l*,
*l'*,*a*,*a'*
give the number of cells that are strictly *north*, *south*,
*east*
and*west* of *s* in µ. The symbol
is used to express that this product omits the corner cell with *l'=a'=0*.

On the surface this formula appears to define only a rational function
of *q* and *t*. Nevertheless, computer data and representation
theoretical

considerations lead us to conjecture that *C_n(q,t)* evaluates
for all *n* to a polynomial with non-negative integer coefficients
and total degree .
In fact, a perusal of the tables in the appendix quickly reveals that the
coefficients of the resulting polynomial always add up to the familiar
Catalan number

By specializing the values of *q* and *t* in the *(q,t)*-Catalan
numbers one also recovers the *q*-Catalan numbers.

The q, t-Catalan numbers arise naturally in the study of diagonal harmonics. They are conjectured to be the coefficient of the sign representation in the Frobenius series for the diagonal harmonics.

The (q,t)-Catalan numbers are now known to be polynomials in q and t
and there exists a proof in the article

Identities and Postitivity Conjectures for some remarkable Operators
in the Theory of Symmetric Functions (to appear), François
Bergeron, A. Garsia, M.
Haiman, G. Tesler

This paper also explains the connections
between the space of diagonal harmonics and the Macdonald polynomials with
the operator Nabla.

The question that these polynomials have non-negative integer coefficients is still unsettled.

Adapeted from the paper:

A REMARKABLE $q,t$-CATALAN SEQUENCE
AND $q$-LAGRANGE INVERSION, Adriano
Garsia, Mark Haiman