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5. Vector Partition Functions and Hyperplanes


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      Problem 5.1.

      Study the quasipolynomial leading coefficients for $2$-row vector partition functions. Find efficient computational methods for finding these coefficients.


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          Problem 5.2.

          Let $A$ be a finite, real, linear hyperplane arrangement with isometric chambers. Is $A$ a Coxeter arrangment?
              For dimensions 2 and 3, the answer is known to be yes.


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              Problem 5.3.

              What is the easiest way to compute the pieces of a vector partition function?
                  This is equivalent to a triangulation problem.


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                  Problem 5.4.

                  onsider the $A_{n-1}$ vector partition function.
                  • How many pieces are there? This is known for $n\leq 7$, and there is known lower bound of $2^{\lfloor \frac{n}{2}\rfloor}$.
                  • Understand why some polynomials factor nicely over the arrangement.
                  • Consider $q$-analogues. In particular, the $q$ version of $(1,2,3,\cdots)$ is conjectured to grow at $e^{\theta(n^2)}$.

                      Cite this as: AimPL: Polyhedral geometry and partition theory, available at http://aimpl.org/polypartition.