First-Derivative Chromatic Symmetric Reconstruction For Proper Trees

Let $T$ be a tree. Stanley asked whether the chromatic symmetric function $X_T$ determines $T$ up to isomorphism. We approach this open problem by regarding $X_T$ as a polynomial in the power-sum symmetric functions $p_1,p_2,\dots$ and studying the invariant ${\mathrm{\Phi }}_T=(\mathrm{\partial }{X}_T\mathrm{/}\mathrm{\partial }{p}_1){\mathrm{\mid }}_{p_1=0}$. We prove that ${\mathrm{\Phi }}_T$ distinguishes every proper tree whose weighted skeleton, the tree obtained from $T$ by weighted contraction of all leaf edges, has distinct weights at non-leaf vertices. We prove further an equivalent formulation of Stanley’s question obtained by attaching a fixed positive number of leaves to every vertex of a tree. Finally, we count spanning forests with at most $t$ edges, grouping them by the sizes of their connected components. We prove that these counts cannot distinguish all trees on $k\ge 4$ vertices unless $t\ge \lfloor k\mathrm{/}2\rfloor$.