Proper circular arc graphs are $e$-positive

Source abstract

We prove an $e$-positive formula for the chromatic symmetric function of proper circular arc graphs solving the $q=1$ case of Ellzey's conjecture. In doing so, we provide a new proof of the $e$-positivity of unit interval graphs, which alongside Guay-Paquet's reduction gives a new proof of the Stanley--Stembridge conjecture. We define color matrices, which count proper colorings, and tableau matrices, whose entries are nonnegative rational numbers and ratios of elementary symmetric functions. We prove the two matrices are related by a single family of change of basis matrices, which become invertible after restricting to finitely many colors, and we show the chromatic symmetric function of proper circular arc graphs comes from taking the trace of these matrices. Using Hikita's tableaux, this gives an explicit formula for the chromatic symmetric function of a proper circular arc graph as a weighted sum over tableaux whose first and last $k$ vertices lie in the same columns.