We define the notions of successive ranks and generalized Durfee squares for overpartitions. We show how these combinatorial statistics give extensions to overpartitions of combinatorial interpretations in terms of lattice paths of the generalizations of the Rogers-Ramanujan identities due to Burge, Andrews and Bressoud. Our result includes the Andrews-Gordon identities, the generalization of the Gordon-Gollnitz identities and Gordon's theorems for overpartitions. This is joint work with Olivier Mallet (Paris 7).

Guo-Niu Han and I have been calculating various multivariable statistical distributions on signed words and permutations of the B and D types, particularly statistical distributions involving the length function  and the record values. My intention is to give an overview on the techniques that have been used. See our three papers on this subject (one to appear, the second published, the third one submitted) that can be downloaded from our home pages.

We propose several formulas refining the enumeration of totally symmetric self-complementary plane partitions (abbreviated as TSSCPP) of size n.

I shall present the generalized non-crossing partitions of Drew Armstrong and the generalized cluster complex of Fomin and Reading. Both of these combinatorial objects are associated to finite reflection groups. They are fascinating in many ways. In particular, they have extremely interesting enumerative properties, some of which I will mention. The main result of the talk will be a surprising relation between the M\"obius function of the poset of generalized non-crossing partitions and certain face numbers of the generalized cluster complex. As yet, there is no intrinsic understanding for this relation, my proof being case-by-case (with, in fact, a gap to be filled in type D).

We exhibit some constant term identities that involve generating functions for "jagged partitions" and describe the partition-theoretic implications. These are in terms of partitions where the number of occurrences of j and j+1 is at most 2. For example, we give interpretations of the square of the product in the first Rogers-Ramanujan identity as well as the product in the Capparelli theorem.

The Painlevé equations are non-linear ordinary differential equations without movable branching points. There are families of special polynomials, such as Yablonskii-Vorob'ev polynomials and Umemura polynomials, associated with rational or algebraic solutions of the Painlevé equations and their generalizations. In this talk we discuss combinatorial properties and positivity of these polynomials.

We will discuss Schur Q-polynomials evaluated on a geometric progression, or equivalently q-enumeration of marked shifted tableaux. We are particularly interested in finding explicit formulas that remain regular for q=1. We have found several such expressions involving determinants and pfaffians of q-ultraspherical polynomials, and also as multiple q-hypergeometric series.

I will present a proof of Frenkel and Turaev's elliptic hypergeometric 10-V-9 summation by enumeration of lattice paths with respect to an elliptic (i.e., doubly periodic meromorphic) weight function. This appears to be the first combinatorial proof of the 10-V-9 summation formula, and at the same time of some important degenerate cases including Jackson's very-well-poised balanced 8-phi-7 and Dougall's 7-F-6 summation. The computation of the elliptic generating function for selected families of nonintersecting lattice paths leads, via the Gessel-Viennot theorem and an elliptic determinant evaluation by Warnaar, to a multivariable extension of the 10-V-9 summation which turns out to be a special case of an identity originally conjectured by Warnaar, subsequently proved by Rosengren. I will also touch on related issues such as elliptic Schur functions and the elliptic enumeration of tableaux and plane partitions.

The cyclic sieving phenomenon (CSP) is an explanation for why q being a root unity gives a recognizable integer in many generating functions. In this talk several known examples of the CSP will be given. Moreover the conjectured enumeration of new plane partition symmetry classes are offered. Connections to invariant theory over finite fields will be discussed.

In this talk I will describe a new class of basic hypergeometric series labelled by the Lie algebra sl_3. The summand of the sl_3 series depends on a pair of Macdonald polynomials and a bisymmetric function related to alternating sign matrices and sl_3 Selberg integrals.