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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asked Oct 10, 2011 at 2:01
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$\begingroup$ Thank you for the leads, lowerbound and F.C. Perhaps via Sturmfels/Thomas' paper and discussions in Strang's book Introduction to Applied Mathematics on optimization and nonlinear programming the connection between Stasheff associahedra as secondary polytopes for Grobner fans/bases for max/min problems and the associahedra's f-vectors as coefficients of compositional inversion through a Lagrange inversion/Legendre transform (OEIS A133437) can be made more explicit. For details on Grobner bases and their secondary polytopes see Rekha Thomas' Lectures on Geometric Combinatorics. $\endgroup$– Tom CopelandCommented Oct 10, 2011 at 21:27
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$\begingroup$ I'm having trouble understanding what is being asked. In the title you ask about the connection between Grobner fans and associahedra which you seem to have figured out (Grobner fans are the normal fans of state polytopes, and these coincide with secondary polytopes at least in the unimodular case). The body asks about the f-vector of associahedra, Lagrange inversion etc, which doesn't seem to have anything to do with Grobner fans... I'm left with the interpretation "why are associahedra ubiquitous?", but it seems like you meant something more specific than that. $\endgroup$– Gjergji ZaimiCommented Oct 10, 2011 at 23:00
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$\begingroup$ See also De Loera, Rambau, and Leal "Triangulations of Point Sets." $\endgroup$– Tom CopelandCommented Oct 10, 2011 at 23:08
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2 Answers
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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
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$\begingroup$ Thank you. Interesting connections. I could follow up on this to find the general answer. $\endgroup$ Commented Oct 11, 2011 at 0:28
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$\begingroup$ The link to
springerlink.comis broken. I'm also unable to find any snapshot saved on the Wayback Machine. $\endgroup$ Commented Dec 17, 2022 at 12:28
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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 n-polygon. 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.
