The Path-Cycle Symmetric Function of a Digraph

Abstract

Recently, Stanley [21] has defined a symmetric function generalization of the chromatic polynomial of a graph. Independently, Chung and Graham [4] have defined a digraph polynomial called the cover polynomial which is closely related to the chromatic polynomial of a graph (in fact, as we shall see, the cover polynomial of a certain digraph associated to a poset P coincides with the chromatic polynomial of the incomparability graph of P) and also to rook polynomials. The starting point for the present paper is the suggestion in Chung and Graham's paper that the cover polynomial might generalize to a symmetric function in much the same way that the chromatic polynomial does. This is indeed the case, and in this paper we shall study this symmetric function generalization of the cover polynomial. As one would expect, there are a number of generalizations and analogues of known results about the cover polynomial, the rook polynomial, and Stanley's chromatic symmetric function. In addition, however, we will obtain some unexpected dividends, such as a combinatorial reciprocity theorem that answers a question of Chung and Graham and ties together a number of known results that previously seemed unrelated ([2, Theorem 7.3], [19, Chapter 7, Theorem 2], [21, Theorem 4.2], [23, Theorem 3.2]), and a new symmetric function basis that appears to be a natural counterpart of the polynomial basis ( x+n d )n=0, ..., d . One reason for these unexpected results is that this topic lies at the intersection of several branches of combinatorics; their interaction naturally gives rise to new connections and ideas. We shall indicate several directions for further research in this potentially very rich area of study.