  In 1995 Stanley conjectured that the chromatic symmetric functions of the
graphs $P_{d,2}$, which we call triangular ladders, were $e$-positive. In this
paper we confirm this conjecture, which is also an unsolved case of the
celebrated $(3+1)$-free conjecture. Our method is to follow the generalization
of the chromatic symmetric functions by Gebhard and Sagan to symmetric
functions in non-commuting variables. These functions satisfy a
deletion-contraction property unlike the chromatic symmetric function in
commuting variables. We do this by proving a new signed combinatorial formula
for \emph{all} unit interval graphs on the basis of elementary symmetric
functions. Then we prove $e$-positivity for triangular ladders by very
carefully defining a sign-reversing involution on our signed combinatorial
formula. This leaves us with certain positive terms and further allows us to
expand on an already-known family of $e$-positive graphs by Gebhard and Sagan.
