Chromatic symmetric functions and change of basis

We prove necessary conditions for certain elementary symmetric functions, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>e</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:math> , to appear with nonzero coefficient in Stanley’s chromatic symmetric function as well as in the generalization considered by Shareshian and Wachs. We do this by first considering the expansion in the monomial or Schur basis and then performing a basis change. Using the former, we make a connection with two fundamental graph theory invariants, the independence and clique numbers. This allows us to prove nonnegativity of three-column coefficients for all natural unit interval graphs, giving more insight into the Stanley–Stembridge Conjecture, recently proven by Hikita, and the Shareshian–Wachs Conjecture. The Schur basis permits us to give a new interpretation of the coefficient of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>e</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> in terms of tableaux. We are also able to give an explicit formula for that coefficient.