  Point-to-point permutation channels are useful models of communication
networks and biological storage mechanisms and have received theoretical
attention in recent years. Propelled by relevant advances in this area, we
analyze the permutation adder multiple-access channel (PAMAC) in this work. In
the PAMAC network model, $d$ senders communicate with a single receiver by
transmitting $p$-ary codewords through an adder multiple-access channel whose
output is subsequently shuffled by a random permutation block. We define a
suitable notion of permutation capacity region $\mathcal{C}_\mathsf{perm}$ for
this model, and establish that $\mathcal{C}_\mathsf{perm}$ is the simplex
consisting of all rate $d$-tuples that sum to $d(p - 1) / 2$ or less. We
achieve this sum-rate by encoding messages as i.i.d. samples from categorical
distributions with carefully chosen parameters, and we derive an inner bound on
$\mathcal{C}_\mathsf{perm}$ by extending the concept of time sharing to the
permutation channel setting. Our proof notably illuminates various connections
between mixed-radix numerical systems and coding schemes for multiple-access
channels. Furthermore, we derive an alternative inner bound on
$\mathcal{C}_\mathsf{perm}$ for the binary PAMAC by analyzing the root
stability of the probability generating function of the adder's output
distribution. Using eigenvalue perturbation results, we obtain error bounds on
the spectrum of the probability generating function's companion matrix,
providing quantitative estimates of decoding performance. Finally, we obtain a
converse bound on $\mathcal{C}_\mathsf{perm}$ matching our achievability
result.
