Adriano M. Garsia

A Remarkable q , t -Catalan Sequence and q -Lagrange Inversion

AbstractWe introduce a rational function Cn(q, t) and conjecture that it always evaluates to a polynomial in q, t with non-negative integer coefficients summing to the familiar Catalan number

An Introduction to Cohen-Macaulay Partially Ordered Sets

\frac{1}{{n + 1}}\left( {\begin{array}{*{20}c} {2n} \\ n \\ \end{array} } \right)

Combinatorial methods in the theory of Cohen-Macaulay rings

. We give supporting evidence by computing the specializations

A New Algorithm for Minimum Cost Binary Trees

D_n \left( q \right) = C_n \left( {q{1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-\nulldelimiterspace} q}} \right)q^{\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)}

A Rogers-Ramanujan bijection

and Cn(q) = Cn(q, 1) = Cn(1,q). We show that, in fact, Dn(q)q -counts Dyck words by the major index and Cn(q) q -counts Dyck paths by area. We also show that Cn(q, t) is the coefficient of the elementary symmetric function en in a symmetric polynomial DHn(x; q, t) which is the conjectured Frobenius characteristic of the module of diagonal harmonic polynomials. On the validity of certain conjectures this yields that Cn(q, t) is the Hilbert series of the diagonal harmonic alternants. It develops that the specialization DHn(x; q, 1) yields a novel and combinatorial way of expressing the solution of the q-Lagrange inversion problem studied by Andrews [2], Garsia [5] and Gessel [11]. Our proofs involve manipulations with the Macdonald basis {Pμ(x; q, t)}μ which are best dealt with in λ-ring notation. In particular we derive here the λ-ring version of several symmetric function identities.

Combinatorics of the Free Lie Algebra and the Symmetric Group

Abstract A descent class, in the symmetric group Sn, is the collection of permutations with a given descent set. It was shown by L. Solomon (J. Algebra41 (1976), 255–264) that the product (in the group algebra Q(Sn)) of two descent classes is a linear combination of descent classes. Thus descent classes generate a subalgebra of Q(Sn). We refer to it here as Solomons descent algebra and denote it by Σn. This algebra is not semisimple but it has a faithul representation in terms of upper triangular matrices. The main goal of this paper is a decomposition of its multiplicative structure. It develops that Σn acts in a natural way on the so-called Lie monomials. This action has a purely combinatorial description and is a crucial tool in the construction of a complete set of indecomposable representations of Σn. In particular we obtain a natural basis of irreducible orthogonal idempotents Σλ (indexed by partitions of n) for the quotient Σ n √Σ n . Natural bases of nilpotents and idempotents for the subspaces EλΣnEμ, for two arbitrary partitions λ and μ, are also constructed and the dimensions of these spaces are given a combinatorial interpretation in terms of the so-called decreasing factorization of an arbitary word into a product of Lyndon words.