  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 $\sum_{i=1}^n r_i\leq \sum_{i=1}^n
t_i, \mbox{ } r_i\in \mathbb{R}_+^n,$ where the $(r_i)_{i\in [n]}$ are
variables. Specifically, we will discuss the central Dahmen-Micchelli space of
the broken wheel graph $BW_n$ and its dual, the $\mathcal{P}$-central space. We
will observe that the $\mathcal{P}$-central space of $BW_n$ is monomial, with a
basis given by the $BW_n$-parking functions. We will show that the volume
polynomial of the the Stanley-Pitman polytope lies in the central
Dahmen-Micchelli space of $BW_n$ and is precisely the polynomial in a
particular basis of the central Dahmen-Micchelli space which corresponds to the
monomial $t_1t_2\cdots t_n$ in the dual monomial basis of the
$\mathcal{P}$-central space. We will then define the generalized broken wheel
graph $GBW_n(T)$ for a given rooted tree $T$ on $n$ vertices. For every such
tree, we can construct $2^{n-1}$ directed graphs, which we will refer to as
\textit{generalized broken wheel graphs}. Each generalized broken wheel graph
constructed from $T$ will give us a polytope, its volume polynomial, and a
\textit{reference monomial}. The $2^{n-1}$ polytopes together give a polyhedral
subdivision of $\mathfrak{Sim}_n(t_1,...,t_n)$, their volume polynomials
together give a basis for the subspace of homogeneous polynomials of degree $n$
of the corresponding central Dahmen-Micchelli space, and their reference
monomials together give a basis for its dual.
