A power sum expansion for the Kromatic symmetric function

The chromatic symmetric XG function is a symmetric function generalization of the chromatic polynomial of a graph, introduced by Stanley (1995). XG is defined by assigning a variable to each color and taking the sum over all proper vertex colorings of G of the product of all variables corresponding to colors used, counted with multiplicity. Stanley (1995) gave an expansion formula for XG in terms of the power sum symmetric functions p_λ using the principle of inclusion-exclusion. Crew, Pechenik, and Spirkl (2023) defined the Kromatic symmetric function X̄G as a K-theoretic analogue of XG, constructed in the same way except that each vertex is assigned a nonempty set of colors such that adjacent vertices have nonoverlapping color sets. They defined a K-analogue p̄_λ of the power sum basis and computed the first few coefficients of the p̄-expansion of X̄G for some small graphs G. They conjectured that the p̄-expansion always has integer coefficients and asked whether there is an explicit formula for these coefficients. We give a formula for the p̄-expansion of X̄G, and prove that the coefficients are indeed always integers.