Abstract. Tropical algebraic geometry is the geometry of the tropical semiring (R, min, +). The theory of total positivity is a natural generalization of the study of matrices with all minors positive. In this paper we introduce the totally positive part of the tropicalization of an arbitrary affine variety, an object which has the structure of a polyhedral fan. We then investigate the case of the Grassmannian, denoting the resulting fan Trop + Gr k,n . We show that Trop + Gr 2,n is the Stanley-Pitman fan, which is combinatorially the fan dual to the (type A n−3 ) associahedron, and that Trop + Gr 3,6 and Trop + Gr 3,7 are closely related to the fans dual to the types D 4 and E 6 associahedra. These results are strikingly reminiscent of the results of Fomin and Zelevinsky, and Scott, who showed that the Grassmannian has a natural cluster algebra structure which is of types A n−3 , D 4 , and E 6 for Gr 2,n , Gr 3,6 , and Gr 3,7 . We suggest a general conjecture about the positive part of the tropicalization of a cluster algebra.

We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. Moreover, we work in the generality of principal coefficients. An immediate corollary of our formulas is a proof of the positivity conjecture of Fomin and Zelevinsky for cluster algebras from surfaces, in geometric type. Contents

In this paper we introduce and study a class of tableaux which we call permutation tableaux; these tableaux are naturally in bijection with permutations, and they are a distinguished subset of the -diagrams of Alex Postnikov [A. Postnikov, Webs in totally positive Grassmann cells, in preparation; L. Williams, Enumeration of totally positive Grassmann cells, Adv. Math. 190 (2005) 319-342]. The structure of these tableaux is in some ways more transparent than the structure of permutations; therefore we believe that permutation tableaux will be useful in furthering the understanding of permutations. We give two bijections from permutation tableaux to permutations. The first bijection carries tableaux statistics to permutation statistics based on relative sizes of pairs of letters in a permutation and their places. We call these statistics weak excedance statistics because of their close relation to weak excedances. The second bijection carries tableaux statistics (via the weak excedance statistics) to statistics based on generalized permutation patterns. We then give enumerative applications of these bijections. One nice consequence of these results is that the polynomial enumerating permutation tableaux according to their content generalizes both Carlitz' q-analog of the Eulerian numbers [L. Carlitz, q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc. 76 (1954) 332-350] and the more recent q-analog of the Eulerian numbers found in [L. Williams, Enumeration of totally positive Grassmann cells, Adv. Math. 190 (2005) 319-342]. We conclude our paper with a list of open problems, as well as remarks on progress on these problems which has been made by A. Burstein, S. Corteel, N. Eriksen, A. Reifegerste, and X. Viennot.

Abstract. We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. Moreover, we work in the generality of principal coefficients. An immediate corollary of our formulas is a proof of the positivity conjecture of Fomin and Zelevinsky for cluster algebras from surfaces, in geometric type.

The partially asymmetric exclusion process (PASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites. It is partially asymmetric in the sense that the probability of hopping left is q times the probability of hopping right. Additionally, particles may enter from the left with probability α and exit from the right with probability β.In this paper we prove a close connection between the PASEP and the combinatorics of permutation tableaux. (These tableaux come indirectly from the totally nonnegative part of the Grassmannian, via work of Postnikov, and were studied in a paper of Steingrímsson and the second author.) Namely, we prove that in the long time limit, the probability that the PASEP is in a particular configuration τ is essentially the generating function for permutation tableaux of shape λ(τ ) enumerated according to three statistics. The proof of this result uses a result of Derrida, Evans, Hakim, and Pasquier on the matrix ansatz for the PASEP model.As an application, we prove some monotonicity results for the PASEP. We also derive some enumerative consequences for permutations enumerated according to various statistics such as weak excedence set, descent set, crossings, and occurrences of generalized patterns.