On the chromatic symmetric function, the generalized degree sequence and the subtree polynomial

R. Stanley asked whether a tree is determined up to isomorphism by its chromatic symmetric function. We approach this question by studying the relationship between the chromatic symmetric function and other invariants. First, we prove Crew’s conjecture that the chromatic symmetric function of a tree determines its generalized degree sequence, which enumerates vertex subsets by cardinality and the numbers of internal and external edges. Second, we prove that the restriction of the generalized degree sequence to subtrees contains exactly the same information as the subtree polynomial, which enumerates subtrees by cardinality and number of leaves. Third, we construct arbitrarily large families of trees sharing the same subtree polynomial, proving and generalizing a conjecture of Eisenstat and Gordon.

This is joint work with Jeremy L. Martin (KU) Jennifer Wagner (Wishburn) and José Zamora (UNAB,Chile)

For information about the Richard P. Stanley Seminar in Combinatorics, visit https://math.mit.edu/combin/