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 X G 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 X G ) 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.