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In this paper we explore the combinatorics of the nonnegative part (G/P )≥0 of a cominuscule Grassmannian. For each such Grassmannian we define Γ -diagrams — certain fillings of generalized Young diagrams which are in bijection with the cells of (G/P )≥0. In the classical cases, we describe Γ -diagrams explicitly in terms of pattern avoidance. We also… (More)

The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f -, hand γvectors. These polytopes include permutohedra, associahedra, graphassociahedra, simple graphic zonotopes, nestohedra, and other interesting polytopes. We give several explicit formulas for h-vectors and γ-vectors involving descent statistics.… (More)

- Sylvie Corteel, Lauren Williams
- Proceedings of the National Academy of Sciences…
- 2010

We introduce some combinatorial objects called staircase tableaux, which have cardinality 4(n)n!, and connect them to both the asymmetric exclusion process (ASEP) and Askey-Wilson polynomials. The ASEP is a model from statistical mechanics introduced in the late 1960s, which describes a system of interacting particles hopping left and right on a… (More)

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… (More)

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… (More)

- Guy Hokanson, Otto Borchert, +11 authors Lauren Williams
- ICALT
- 2008

- Rachel N. Caskey, Jeffrey Zaman, +7 authors Andrew Dallas Boyd
- Pediatrics
- 2014

BACKGROUND AND OBJECTIVES
Diagnostic codes are used widely within health care for billing, quality assessment, and to measure clinical outcomes. The US health care system will transition to the International Classification of Diseases, 10th Revision, Clinical Modification (ICD-10-CM), in October 2015. Little is known about how this transition will affect… (More)

We construct two bases for each cluster algebra coming from a triangulated surface without punctures. We work in the context of a coefficient system coming from a full-rank exchange matrix, for example, principal coefficients.

- Lauren Williams
- 2005

Tropical varieties play an important role in algebraic geometry. The Bergman complex B(M) and the positive Bergman complex B+(M) of an oriented matroid M generalize to matroids the notions of the tropical variety and positive tropical variety associated to a linear ideal. Our main result is that if A is a Coxeter arrangement of type Φ with corresponding… (More)

In this paper we use toric geometry to investigate the topology of the totally non-negative part of the Grassmannian, denoted (Grk,n)≥0. This is a cell complex whose cells G can be parameterized in terms of the combinatorics of plane-bipartite graphs G. To each cell G we associate a certain polytope P(G). The polytopes P(G) are analogous to the well-known… (More)