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 present 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. The precise form of the conjecture is given in the J. where it is further conjectured that C n (q, t) is the Hilbert Series of the… (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)
This work is gratefully dedicated to Dominique Foata for his inspiring and pioneering work in algebraic combinatorics. We hope that he will nd it to be in harmony with the Lotharingian spirit which he has nurtured for so many years. ABSTRACT. We construct for each`n a bigraded S n-module H and conjecture that its Frobenius characteristic C (x; q; t) yields… (More)
The dissertation of Sally Picciotto is approved, and it is acceptable in quality and form for publication on microolm: who have witnessed and greatly assisted in my growth over the last few years.
These notes cover the contents of a series of lectures in a Topics in Algebraic Combinatorics course given at UCSD in Winter 2001. The initial effort was prompted by a desire to understand the connections between the theory of reduced decompositions started by the pioneering paper  of R. Stanley and the theory of balanced tabloids studied by C. Green et… (More)
Astract We study here the ring QS n of Quasi-Symmetric Functions in the variables x 1 , x 2 ,. .. , x n. F. Bergeron and C. Reutenauer  formulated a number of conjectures about this ring, in particular they conjectured that it is free over the ring Λ n of symmetric functions in x 1 , x 2 ,. .. , x n. We present here an algorithm that recursively… (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)