PubMed Health⌕ Search

Biomedical subjects

A M Garsia

Publications and source records attributed to A M Garsia.

4 recordsLinked to original sources
PubMed2001

A positivity result in the theory of Macdonald polynomials.

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 collection (n) of Dyck paths Pi of length 2n yielding C(n)(q, t) = summation operator(pi) t(a(Pi))q(b(Pi)). Our proof is based on a recursion for C(n)(q, t) suggested by a pair of statistics recently proposed by J. Haglund. One of the byproducts of our results is a proof of the validity of Haglund's conjecture.

Journal Article↗
PubMed1993

A graded representation model for Macdonald's polynomials.

We define doubly graded Sn modules Rmu for which we conjecture that the multiplicities of irreducible representations in various bi-degrees are given by the Macdonald coefficients Klambdamu. Assuming one fundamental conjecture, the modules Rmu can be given several equivalent definitions, which we discuss. We prove the conjectures in various special cases.

Journal Article↗
PubMed1981

Method for constructing bijections for classical partition identities.

We sketch the construction of a bijection between the partitions of n with parts congruent to 1 or 4 (mod 5) and the partitions of n with parts differing by at least 2. This bijection is obtained by a cut-and-paste procedure that starts with a partition in one class and ends with a partition in the other class. The whole construction is a combination of a bijection discovered quite early by Schur and two bijections of our own. A basic principle concerning pairs of involutions provides the key for connecting all these bijections. It appears that our methods lead to an algorithm for constructing bijections for other identities of Rogers-Ramanujan type such as the Gordon identities.

Journal Article↗