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- A M Garsia, M Haiman
- 1996

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)

- F Bergeron, N Bergeron, A M Garsia, M Haiman, G Tesler
- 1998

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 M Garsia, M Haiman
- 1996

This work is gratefully dedicated to Dominique Foata for his inspiring and pioneering work in algebraic combinatorics. We hope that he will find it to be in harmony with the Lotharingian spirit which he has nurtured for so many years. ABSTRACT. We construct for each µ n a bigraded S n-module H µ and conjecture that its Frobenius characteristic C µ (x; q, t)… (More)

- A M Garsia, C Reutenauer
- 2003

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)

- A M Garsia, M Haiman, G Tesler
- 2000

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)

- F Bergeron, A M Garsia
- 1999

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)

- F Bergeron, A M Garsia
- 1998

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)

- A Garsia, J Haglund, A M Garsia
- 2008

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)