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The machinery of zonotopal algebra is linked with two particular polytopes:the Stanley-Pitman polytope and the regular simplex$\mathfrak{Sim}_n(t_1,...,t_n)$ with parameters $t_1,...,t_n\in\mathbb{R}_+^n$, defined by the inequalities $\sum_{i=1}^n r_i\leq \sum_{i=1}^nt_i, \mbox{ } r_i\in \mathbb{R}_+^n,$ where the $(r_i)_{i\in [n]}$ arevariables. Specifically, we will discuss the central Dahmen-Micchelli space ofthe broken wheel graph $BW_n$ and its dual, the $\mathcal{P}$-central space. Wewill observe that the $\mathcal{P}$-central space of $BW_n$ is monomial, with abasis given by the $BW_n$-parking functions. We will show that the volumepolynomial of the the Stanley-Pitman polytope lies in the centralDahmen-Micchelli space of $BW_n$ and is precisely the polynomial in aparticular basis of the central Dahmen-Micchelli space which corresponds to themonomial $t_1t_2\cdots t_n$ in the dual monomial basis of the$\mathcal{P}$-central space. We will then define the generalized broken wheelgraph $GBW_n(T)$ for a given rooted tree $T$ on $n$ vertices. For every suchtree, we can construct $2^{n-1}$ directed graphs, which we will refer to as\textit{generalized broken wheel graphs}. Each generalized broken wheel graphconstructed from $T$ will give us a polytope, its volume polynomial, and a\textit{reference monomial}. The $2^{n-1}$ polytopes together give a polyhedralsubdivision of $\mathfrak{Sim}_n(t_1,...,t_n)$, their volume polynomialstogether give a basis for the subspace of homogeneous polynomials of degree $n$of the corresponding central Dahmen-Micchelli space, and their referencemonomials together give a basis for its dual.

The Zonotopal Algebra of the Broken Wheel Graph and its Generalization