UBC Discrete Math Seminar: Shamil Asgarli

The chromatic polynomial of a graph G counts the number of proper k-colorings of G. In 1995, Stanley extended this concept to define the chromatic symmetric function of a graph, which lives in the ring Sym of all symmetric functions. Building on this work, Shareshian and Wachs introduced the chromatic quasisymmetric function (CQF) in 2016, a quasisymmetric invariant associated with labeled graphs. The CQF of a labeled graph G lives in the polynomial ring QSym[q], where q is a formal variable, and QSym denotes the ring of quasisymmetric functions in infinitely many variables. A natural question arises: when does the CQF of G live in the smaller ring Sym[q]? We completely answer this question for labeled path graphs. Our main result establishes that the CQF of any labeled P_n is not symmetric, except when the labeling is either the natural ordering 1, 2, ...., n or its reverse n, n-1, ..., 1 (the two labelings are isomorphic). This is joint work with Farid Aliniaeifard, Maria Esipova, Ethan Shelburne, Stephanie van Willigenburg, Tamsen Whitehead McGinley.