A polynomial associated with rooted trees and specific posets

TL;DR

This paper introduces a new polynomial for rooted trees and certain posets, revealing its recursive properties, combinatorial interpretations, and characterizations of tree isomorphism, along with a generalization to a broader class of posets.

Contribution

It extends a polynomial invariant for rooted trees, explores its properties, and generalizes it to a new class of posets called $\\mathcal{V}$-posets, with implications for tree characterization.

Findings

The polynomial satisfies a deletion-contraction recursion.

Two bivariate specializations uniquely identify trees up to isomorphism.

The polynomial can be generalized to $\\mathcal{V}$-posets.

Abstract

We investigate a trivariate polynomial associated with rooted trees. It generalises a bivariate polynomial for rooted trees that was recently introduced by Liu. We show that this polynomial satisfies a deletion-contraction recursion and can be expressed as a sum over maximal antichains. Several combinatorial quantities can be obtained as special values, in particular the number of antichains, maximal antichains and cutsets. We prove that two of the three possible bivariate specialisations characterise trees uniquely up to isomorphism. One of these has already been established by Liu, the other is new. For the third specialisation, we construct non-isomorphic trees with the same associated polynomial. We finally find that our polynomial can be generalised in a natural way to a family of posets that we call $\mathcal{V}$-posets. These posets are obtained recursively by either disjoint unions or adding a greatest/least element to existing $\mathcal{V}$-posets.