We present a proof of a Littlewood-Richardson rule for the K-theory of odd
orthogonal Grassmannians OG(n,2n+1), as conjectured in [Thomas-Yong '09].
Specifically, we prove that rectification using the jeu de taquin for
increasing shifted tableaux introduced there, is well-defined and gives rise to
an associative product. Recently, [Buch-Ravikumar '09] proved a Pieri rule for
OG(n,2n+1) that [Feigenbaum-Sergel '09] showed confirms a special case of the
conjecture. Together, these results imply the aforementioned conjecture.

Higher Auslander algebras were introduced by Iyama generalizing classical
concepts from representation theory of finite dimensional algebras. Recently
these higher analogues of classical representation theory have been
increasingly studied. Cyclic polytopes are classical objects of study in convex
geometry. In particular, their triangulations have been studied with a view
towards generalizing the rich combinatorial structure of triangulations of
polygons. In this paper, we demonstrate a connection between these two
seemingly unrelated subjects.

We propose a definition of an "oriented interval greedoid" that
simultaneously generalizes the notion of an oriented matroid and the
construction on antimatroids introduced by L. J. Billera, S. K. Hsiao, and J.
S. Provan in "Enumeration in convex geometries and associated polytopal
subdivisions of spheres" [Discrete Comput. Geom. 39 (2008), no. 1-3, 123--137].
As for of oriented matroids, associated to each oriented interval greedoid is a
spherical simplicial complex whose face enumeration depends only on the
underlying interval greedoid.