2024/11/28 by Liu, Ricky Ini, Tang, Michael · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2411.18972
The generalized degree polynomial GT(x,y,z) of a tree T is an invariant introduced by Crew that enumerates subsets of vertices by size and number of internal and boundary edges. Aliste-Prieto et al. proved that GT is determined linearly by the chromatic symmetric function XT, introduced by Stanley. We present several classes of information about T that can be recovered from GT and hence also from XT. Examples of such information include the double-degree sequence of T, which enumerates edges of T by the pair of degrees of their endpoints, and the leaf adjacency sequence of T, which enumerates vertices of T by degree and number of adjacent leaves. We also discuss a further generalization of GT that enumerates tuples of vertex sets and show that this is also determined by XT.