  A 1d random geometric graph (1d RGG) is built by joining a random sample of
$n$ points from an interval of the real line with probability $p$. We count the
number of $k$-hop paths between two vertices of the graph in the case where the
space is the 1d interval $[0,1]$. We show how the $k$-hop path count between
two vertices at Euclidean distance $|x-y|$ is in bijection with the volume
enclosed by a uniformly random $d$-dimensional lattice path joining the corners
of a $(k-1)$-dimensional hyperrectangular lattice. We are able to provide the
probability generating function and distribution of this $k$-hop path count as
a sum over lattice paths, incorporating the idea of restricted integer
partitions with limited number of parts. We therefore demonstrate and describe
an important link between spatial random graphs, and lattice path
combinatorics, where the $d$-dimensional lattice paths correspond to spatial
permutations of the geometric points on the line.
