Extended chromatic symmetric functions and equality of ribbon Schur functions

A Vertex-Weighted Tutte Symmetric Function, and Constructing Graphs with Equal Chromatic Symmetric Function

“…In an independent work [2], Aliniaeifard, Wang and van Willigenburg obtained results similar to the ones presented in this section, but written completely in the language of symmetric functions. A 2-pointed vertex-weighted graph is a tuple (G, w, s, t), where (G, w) is a vertexweighted graph, and s and t are (possibly the same) vertices of G. When w, s, and t are clear, we will often just write G in a slight abuse of notation.…”

“…In Section 7, we further use the equivalence between XB G and the W -polynomial of G to find additional families of vertex-weighted graphs with the same XB, and in particular we show how to construct arbitrarily large vertex-weighted paths with equal XB (similar results are found in the independent work [2] by Aliniaeifard, Wang, and van Willigenburg).…”

“…It follows that the homomorphism U of Λ defined by sending each p λ to h λ and then extending linearly transforms the chromatic symmetric function of the weighted path P β into the the ribbon Schur function associated with β. This observation is implicit in [4] and the morphism U is studied in detail in [2].…”

“…The first version of this article did not contain the results of Section 7. While working on this version of the article, the authors became aware of the independent work [2].…”

A Graph Polynomial from Chromatic Symmetric Functions

“…In recent work [6], Spirkl and the second author extended X G to vertex-weighted graphs to give the function a deletion-contraction relation, and Aliniaeifard, Wang, and van Willigenburg [1] showed that extended chromatic symmetric functions of weighted paths correspond to ribbon Schur functions, while also showing that these functions form a basis for the space Λ of all symmetric functions. They note that this path basis, as well as the analogous star basis, are the only bases that satisfy a chromatic reciprocity relation with respect to the power-sum basis.…”