In "Computing Grobner Fans" by Fukuda/Jensen/Thomas on page 2210 in Table 1 are the numbers (1,20,120,300,330,132) for some statistics on Grobner fans for Grass(2,5). This is a vector found in A126216, A033282, and A133437 of http://oeis.org/ related to Stasheff associahedra (also to Lagrange inversion, Dyck paths, and other combinatorial constructs). Is there a more general connection?
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This paper is about Grobner fans and uses a Stasheff associahedron in an example. http://www.springerlink.com/content/y102175023224321/fulltext.pdf 


The Grassmannian Gr(2,n) is closely related to the associahedra, by the mean of Fomin and Zelevinsky's theory of cluster algebras. The natural cluster algebra structure on the space of homogeneous functions on Gr(2,n) can be described using triangulations of an npolygon. Therefore its combinatorics is described by the associahedra. I do not known the role of the Grobner fan in this context. Maybe the works on the tropical Grassmannian could shed some light. 

