Combinatorial Theory

Tatsuyki Hikita recently proved the Stanley-Stembridge conjecture, showing that the \(e\)-coefficients of the chromatic symmetric function of an incomparability graph of a \((\mathbf{3}+\mathbf{1})\) -free poset are non-negative. It remains an open problem to find combinatorial interpretations of the \(e\)-coefficients. For a \((\mathbf{3}+\mathbf{1})\)-free \(P\), we define a hybrid of \(P\)-tableaux and cylindric tableaux called cylindric \(P\)-tableaux. The weight generating function of cylindric \(P\)-tableaux of shape \(\lambda/\mu/d\) are shown to be \(P\)-analogs of cylindric Schur functions defined by a determinantal formula. We deduce that certain sums of the \(e\)-expansion coefficients of the chromatic symmetric function \(X_{\operatorname{inc}(P)}\) are counted by the number of standard cylindric \(P\)-tableaux of the appropriate shape. We connect the \(P\)-analogs of symmetric functions to a theorem on Hecke algebra immanants due to Clearman-Hyatt-Shelton-Skandera.

Mathematics Subject Classifications: 05E05

Keywords: Symmetric Functions, \(e\)-Positivity