标题： 扩展色对称函数和带形施尔函数的相等性 显示英文标题

标题： Extended chromatic symmetric functions and equality of ribbon Schur functions

摘要： 我们证明了加权图的扩展色对称函数的一个一般包含-排除关系，该关系专门化为（扩展）$k$-删除，并且我们给出了两种方法，从加权图中获得对称函数代数的众多新基底。 此外，我们通过证明两个加权路径具有相等的扩展色对称函数当且仅当它们的条带施罗德函数相等，从而对这种情况进行了分类。 后者分类依赖于组合的组合操作，我们将这一操作推广到图的组合。 然后，我们应用我们的推广，以获得无限多个加权图族，其成员具有相等的扩展色对称函数。 显示英文摘要

摘要： We prove a general inclusion-exclusion relation for the extended chromatic symmetric function of a weighted graph, which specializes to (extended) $k$-deletion, and we give two methods to obtain numerous new bases from weighted graphs for the algebra of symmetric functions. Moreover, we classify when two weighted paths have equal extended chromatic symmetric functions by proving this is equivalent to the classification of equal ribbon Schur functions. This latter classification is dependent on the operation composition of compositions, which we generalize to composition of graphs. We then apply our generalization to obtain infinitely many families of weighted graphs whose members have equal extended chromatic symmetric functions.