TITLE: Algorithm to list all Kostant partitions
QUESTION [6 upvotes]: Let $\Phi_+$ be the set of positive roots in some root system, and let $Q_+$ be the positive part of the root lattice, i.e., the set of elements of the form $\sum_{\beta\in \Phi_+}m_\beta\beta$ with $m_\beta \in \mathbb{Z}_{\geq 0}$.
For $\theta\in Q_+$, a Kostant partition of $\theta$ is a sequence $(\beta_1^{(m_1)}, \ldots, \beta_t^{(m_t)})$ such that $\beta_i\in \Phi_+$, $m_i\in \mathbb{Z}_{>0}$, $m_1\beta_1 + \cdots + m_t\beta_t = \theta$, and $\beta_1 > \cdots > \beta_t$ according to some fixed total order on $\Phi_+$.
For example, for the root system of type $A_3$ with simple roots $\alpha_1, \alpha_2, \alpha_3$, the Kostant partitions of $\theta = \alpha_1+2\alpha_2+\alpha_3$ are
$$(\alpha_1+\alpha_2+\alpha_3, \alpha_2), \ (\alpha_2+\alpha_3, \alpha_1+\alpha_2), \ (\alpha_3, \alpha_2, \alpha_1+\alpha_2), \ (\alpha_2+\alpha_3, \alpha_1, \alpha_1), \ (\alpha_3, \alpha_2^{(2)}, \alpha_1).$$
Is there an algorithm for listing or iterating through all Kostant partitions of a given $\theta\in Q_+$ (for an arbitrary root system)?
An inefficient/recursive attempt is as follows:
Phi := the set of positive roots
Q := the positive part of the root lattice

def KostantPartitions(theta):
    L = []
    for beta in Phi:
        if theta-beta is in Q:
            for kp in KostantPartitions(theta-beta):
                new = kp with beta inserted
                if new not in L:
                    L += [new]
    return L

Is there a more efficient algorithm? I realize this question is ill-posed (what does "efficient" mean?); I am really just looking for something better than the above.
I would also be happy with partial answers, e.g., an algorithm for root systems of type $A$.

REPLY [3 votes]: For fixed $\theta$, the set of $(m_{\beta})$ giving a Kostant partition is evidently the set of lattice points in some polyhedron given in terms of hyperplane inequalities and equalities: they are exactly the integer points satisfying the inequalities $m_{\beta} \geq 0$ and equalities coming from $\theta = \sum_{\beta} m_{\beta} \beta$. There are a lot of software out there which can solve the general problem of finding all the lattice points of a given polyhedron: e.g., general math software like Sage can do this (see the documentation at http://doc.sagemath.org/html/en/reference/discrete_geometry/sage/geometry/lattice_polytope.html), but also there is specialized software like "LattE" (https://www.math.ucdavis.edu/~latte/).
Incidentally, this kind of reasoning is what lets you conclude that the Kostant partition function (i.e., the number of Kostant partitions) is a piecewise quasipolynomial, a fact which you may be interested in if you were not aware of it already. And in the particular case of the Type A root system, which you mentioned you were specially interested in, unimodularity implies that the Kostant partition function is a piecewise polynomial; see https://www.sciencedirect.com/science/article/pii/S0021869304000055.