  In this note we present a formula for the Cachazo-Early-Guevara-Mizera (CEGM)
generalized biadjoint amplitudes for all $k$ and $n$ on what we call the
minimal kinematics. We prove that on the minimal kinematics, the scattering
equations on the configuration space of $n$ points on $\mathbb{CP}^{k-1}$ has a
unique solution, and that this solution is in the image of a Veronese
embedding. The minimal kinematics is an all $k$ generalization of the one
recently introduced by Early for $k=2$ and uses a choice of cyclic ordering. We
conjecture an explicit formula for $m_n^{(k)}(\mathbb{I},\mathbb{I})$ which we
have checked analytically through $n=10$ for all $k$. The answer is a simple
rational function which has only simple poles; the poles have the combinatorial
structure of the circulant graph ${\rm C}_n^{(1,2,\dots, k-2)}$. Generalized
biadjoint amplitudes can also be evaluated using the positive tropical
Grassmannian ${\rm Tr}^+{\rm G}(k,n)$ in terms of generalized planar Feynman
diagrams. We find perfect agreement between both definitions for all cases
where the latter is known in the literature. In particular, this gives the
first strong consistency check on the $90\,608$ planar arrays for ${\rm
Tr}^+{\rm G}(4,8)$ recently computed by Cachazo, Guevara, Umbert and Zhang. We
also introduce another class of special kinematics called planar-basis
kinematics which generalizes the one introduced by Cachazo, He and Yuan for
$k=2$ and uses the planar basis recently introduced by Early for all $k$. Based
on numerical computations through $n=8$ for all $k$, we conjecture that on the
planar-basis kinematics $m_n^{(k)}(\mathbb{I},\mathbb{I})$ evaluates to the
multidimensional Catalan numbers, suggesting the possibility of novel
combinatorial interpretations. For $k=2$ these are the standard Catalan
numbers.
