We study the statistics area, bounce and dinv on the set of parallelogram polyominoes having a rectangular m times n bounding box. We show that the bi-statistics (area, bounce) and (area, dinv) give rise to the same q, t-analogue of Narayana numbers which was introduced by two of the authors in [4]. We prove the main conjectures of that paper: the q, t-Narayana polynomials are symmetric in both q and t, and m and n. This is accomplished by providing a symmetric functions interpretation of the q, t-Narayana polynomials which relates them to the famous diagonal harmonics.

We explore the link between combinatorics and probability generated by the question "What does a random parking function look like?" This gives rise to novel probabilistic interpretations of some elegant, known generating functions. It leads to new combinatorics: how many parking functions begin with i? We classify features (e.g., the full descent pattern) of parking functions that have exactly the same distribution among parking functions as among all functions. Finally, we develop the link between parking functions and Brownian excursion theory to give examples where the two ensembles differ.

Let H(n) be the group of 3 × 3 uni-uppertriangular matrices with entries in Z/nZ, the integers mod n. We show that the simple random walk converges to the uniform distribution in order n 2 steps. The argument uses Fourier analysis and is surprisingly challenging. It introduces novel techniques for bounding the spectrum which are useful for a variety of walks on a variety of groups.

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Abstract. In analyzing a simple random walk on the Heisenberg group we encounter the problem of bounding the extreme eigenvalues of an n × n matrix of the form M = C + D where C is a circulant and D a diagonal matrix. The discrete Schrödinger operators are an interesting special case. The Weyl and Horn bounds are not useful here. This paper develops three different approaches to getting good bounds. The first uses the geometry of the eigenspaces of C and D, applying a discrete version of the uncertainty principle. The second shows that, in a useful limit, the matrix M tends to the harmonic oscillator on L 2 (R) and the known eigenstructure can be transferred back. The third approach is purely probabilistic, extending M to an absorbing Markov chain and using hitting time arguments to bound the Dirichlet eigenvalues. The approaches allow generalization to other walks on other groups.

It was conjectured in [5] and proved by Mark Haiman in [13] that the Frobenius Characteristic of the S n Module of Diagonal Harmonics is none other than ∇e n . Here ∇ is the symmetric function operator introduced in [1] with eigen-functions the modified Macdonald basis {H μ } μ . The Shuffle Conjecture [12] expresses the scalar product ∇e n , h μ1 h μ2 • • • h μ k as a weighted sum of Parking Functions on the n × n lattice square which are shuffles of k increasing words. In [10] Jim Haglund succeeded in proving the k = 2 case of this conjecture. In this paper we give a new and more direct proof of the combinatorial part of Haglund's argument and obtain a substantial reduction in the complexity of the symmetric function part. * Work carried out under NSF support.

Recent results have placed the classical shuffle conjecture of Haglund et al. in a broader context of an infinite family of conjectures about parking functions in any rectangular lattice. The combinatorial side of the new conjectures has been defined using a complicated generalization of the dinv statistic which is composed of three parts and which is not obviously nonnegative. Here we simplify the definition of dinv, prove that it is always non-negative, and give a geometric description of the statistic in the style of the classical case. We go on to show that in the (n − 1) × n lattice, parking functions satisfy a fermionic formula that is similar to the one given in the classical case by Haglund and Loehr.