标题： 图的色 MacMahon 对称函数 显示英文标题

标题： Chromatic MacMahon symmetric functions of graphs

摘要： 一个MacMahon对称函数是关于对称群在多个变量字母表的幂级数上的对角作用的不变量。 我们引入了顶点加权图的色数对称函数的类似物，其取值为两个变量集上的MacMahon对称函数，记录有关顶点集的基数和权重的信息。 我们证明了树的色数MacMahon函数通过基数、权重以及内部和外部边的数量确定其顶点子集的生成函数。 这个结果推广了无权重情况下的结果，该结果最初由Crew提出猜想，并由Aliste-Prieto--Martin--Wagner--Zamora和Liu--Tang分别证明。 显示英文摘要

摘要： A MacMahon symmetric function is an invariant of the diagonal action of the symmetric group on power series in multiple alphabets of variables. We introduce an analogue of the chromatic symmetric function for vertex-weighted graphs, taking values in the MacMahon symmetric functions on two sets of variables, recording information about both cardinalities and weights of vertex sets. We prove that the chromatic symmetric MacMahon function of a tree determines the generating function for its vertex subsets by cardinality, weight, and the numbers of internal and external edges. This result generalizes the one for the unweighted case, first conjectured by Crew and proved independently by Aliste-Prieto--Martin--Wagner--Zamora and Liu--Tang.