  We study a random circuit model of constrained fracton dynamics, in which
particles on a one-dimensional lattice undergo random local motion subject to
both charge and dipole moment conservation. The configuration space of this
system exhibits a continuous phase transition between a weakly fragmented
("thermalizing") phase and a strongly fragmented ("nonthermalizing") phase as a
function of the number density of particles. Here, by mapping to two different
problems in combinatorics, we identify an exact solution for the critical
density $n_c$. Specifically, when evolution proceeds by operators that act on
$\ell$ contiguous sites, the critical density is given by $n_c = 1/(\ell -2)$.
We identify the critical scaling near the transition, and we show that there is
a universal value of the correlation length exponent $\nu = 2$. We confirm our
theoretical results with numeric simulations. In the thermalizing phase the
dynamical exponent is subdiffusive: $z=4$, while at the critical point it
increases to $z_c \gtrsim 6$.
