Reconstructing Rooted Trees From Their Strict Order Quasisymmetric Functions

TL;DR

This paper presents a method to reconstruct rooted trees from their strict order quasisymmetric functions, providing a combinatorial proof and identifying key terms that distinguish different rooted trees.

Contribution

It introduces an explicit reconstruction procedure from the strict order quasisymmetric function, advancing understanding of tree invariants in graph theory.

Findings

Reconstruction of rooted trees from their functions is possible with finite sampling.

The method provides a combinatorial proof of previous distinguishability results.

Key terms in the functions uniquely identify rooted trees.

Abstract

Determining whether two graphs are isomorphic is an important and difficult problem in graph theory. One way to make progress towards this problem is by finding and studying graph invariants that distinguish large classes of graphs. Stanley conjectured that his chromatic symmetric function distinguishes all trees, which has remained unresolved. Recently, Hasebe and Tsujie introduced an analogue of Stanley's function for posets, called the strict order quasisymmetric function, and proved that it distinguishes all rooted trees. In this paper, we devise a procedure to explicitly reconstruct a rooted tree from its strict order quasisymmetric function by sampling a finite number of terms. The procedure not only provides a combinatorial proof of the result of Hasebe and Tsujie, but also tracks down the representative terms of each rooted tree that distinguish it from other rooted trees.