Adriano Garsia

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We introduce a rational function C n (q; t) and conjecture that it always evaluates to a polynomial in q; t with non-negative integer coeecients summing to the familiar Catalan number 1 n+1 ? 2n n. We give supporting evidence by computing the spe-cializations D n (q) = C n (q; 1=q) q (n 2) and C n (q) = C n (q; 1) = C n (1; q). We show that, in fact, D n(More)
The lattice cell in the i + 1 st row and j + 1 st column of the positive quadrant of the plane is denoted (i; j). If is a partition of n + 1, we denote by =ij the diagram obtained by removing the cell (i; j) from the (French) Ferrers diagram of. We set =ij = det k x pj i y qj i k n i;j=1 , where (p 1 ; q 1); : : :; (p n ; q n) are the cells of =ij, and let(More)
We outline here a proof that a certain rational function C(n)(q, t), which has come to be known as the "q, t-Catalan," is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. Because C(n)(q, t) evaluates to the Catalan number at t = q = 1, it has also been an open problem to find a pair of statistics a, b on the(More)
A descent class, in the symmetric group S,, is the collection of permutations with a given descent set. It was shown by L. Solomon (J. Algebra 41 (1976), 255-264) that the product (in the group algebra Q(S,)) of two descent classes is a linear combination of descent classes. Thus descent classes generate a subalgebra of Q(.S,). We refer to it here as(More)
For a partition µ = (µ1 > µ2 > · · · > µ k > 0) set Bµ(q, t) = k i=1 t i−1 (1+· · ·+q µ i −1). In [8] Garsia-Tesler proved that if γ is a partition of k and λ = (n−k, γ) is a partition of n, then there is a unique symmetric polynomial kγ(x; q, t) of degree ≤ k with the property that˜K λµ (q, t) = kγ[Bµ(q, t); q, t] holds true for all partitions µ. It was(More)
This work studies the remarkable relationships that hold among certain m-tuples of the Garsia-Haiman modules M µ and corresponding elements of the Macdonald basis. We recall that in [10], M µ is defined for a partition µ ⊢ n, as the linear span of derivatives of a certain bihomogeneous polynomial ∆ µ y n. It has been conjectured in [6], [10] that M µ has n!(More)
This work studies the remarkable relationships that hold among certain m-tuples of the Garsia-Haiman modules M and corresponding elements of the Macdonald basis. We recall that in 10], M is deened for a partition`n, as the linear span of derivatives of a certain bihomogeneouspolynomial (x; y) in the variables x 1 ; x 2 conjecturedin 6], 10] that M has n!(More)
The bigraded Frobenius characteristic of the Garsia-Haiman module M μ is known [7] [10] to be given by the modified Macdonald polynomial˜H μ [X; q, t]. It follows from this that, for μ n the symmetric polynomial ∂ p1˜H μ [X; q, t] is the bigraded Frobenius characteristic of the restriction of M μ from S n to S n−1. The theory of Macdonald polynomials gives(More)