Lauren Williams

Learn More
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)
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)
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)
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)