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## 4. Polytopes and Lattice Point Enumeration

Polytope problems which do not arise from $s$-lecture hall paritions.
1. #### Problem 4.1.

Investigate the volume of flow polytopes by way of geometry. There are many interesting known results such as ${\rm Vol}(K_{n+1}(1,0,0,\cdots,0,-1))=\prod C_i$, a product of Catalan numbers, and the flow polytope of type $D_n$ evaluated at $(2,0,0,\cdots,0)$ is $2^{(n-2)^2}\cdot \prod C_i$ (the same Catalan product). Is there a geometric method (e.g. a coning operation) for obtaining these results? Is there a way of getting a general formula?
• #### Problem 4.2.

Consider the following exponential generating function identity $\left(\sum_{n=0}^\infty(-1)^nh_n\frac{z^n}{n!} \right)^{-1}=\sum_{k=0}^{\infty}\sum_{\pi\in S_k}e_{\lambda(\pi)}\frac{z^n}{n!}$ where $h_n$ denotes the complete homogeneous symmetric functions, and for a parition $\mu=(\mu_1,\cdots, \mu_j)$, $e_\mu=e_{\mu_1}\cdots e_{\mu_j}$ where $e_i$ denotes the elementary symmetric functions.

The specialization $e_i\mapsto t$ gives the Eulerian polynomial. Are there proofs of this identity or other versions using lattice point enumeration?

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