Principal specializations of the $t$-chromatic symmetric function

Source abstract

We introduce the $t$-chromatic symmetric function $Z_G(\mathbf{x};t)$, a variant of Stanley's chromatic symmetric function in which the variable $t$ records the number of distinct colors used in a proper coloring. Although $Z_G(\mathbf{x};t)$ is equivalent to the chromatic symmetric function as a graph invariant, its principal specializations retain substantially more information than those of the chromatic symmetric function. This construction is motivated by the work of Bajo--Beck--Vindas-Meléndez on the $q$-chromatic polynomial and the conjecture of Loehr--Warrington that the $n$th principal specialization of the chromatic symmetric function distinguishes trees on at most $n$ vertices, both of which stem from Stanley's longstanding question of whether the chromatic symmetric function distinguishes trees. We determine the classes of caterpillars resulting from the second principal specialization of $Z_G(\mathbf{x};t)$, showing that the second specialization already captures some of the structure of trees. We then prove that the third principal specialization distinguishes and explicitly reconstructs caterpillars with three spine vertices as well as elongated bistars. Finally, we conjecture, based on computational data, that the fourth principal specialization distinguishes all caterpillars.