Triangular Ladders $P_{d,2}$ are $e$-positive

Abstract

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.