Descents, Quasi-Symmetric Functions, Robinson-Schensted for Posets, and the Chromatic Symmetric Function

Abstract

We investigate an apparent hodgepodge of topics: a Robinson-Schensted algorithm for (3 + 1)-free posets, Chung and Graham's G-descent expansion of the chromatic polynomial, a quasi-symmetric expansion of the path-cycle symmetric function, and an expansion of Stanley's chromatic symmetric function XG in terms of a new symmetric function basis. We show how the theory of P-partitions (in particular, Stanley's quasi-symmetric function expansion of the chromatic symmetric function XG) unifies them all, subsuming two old results and implying two new ones. Perhaps our most interesting result relates to the still-open problem of finding a Robinson-Schensted algorithm for (3 + 1)-free posets. (Magid has announced a solution but it appears to be incorrect.) We show that such an algorithm ought to "respect descents," and that the best partial algorithm so far - due to Sundquist, Wagner, and West - respects descents if it avoids a certain induced subposet.