Positivity among P-partition generating functions

Abstract

We seek simple conditions on a pair of labeled posets that determine when the difference of their $(P,\ddot{I}‰)$-partition enumerators is $F$-positive, i.e., positive in Gessel's fundamental basis. This is a quasisymmetric analogue of the extensively studied problem of finding conditions on a pair of skew shapes that determine when the difference of their skew Schur functions is Schur-positive. We determine necessary conditions and separate sufficient conditions for $F$-positivity, and show that a broad operation for combining posets preserves positivity properties. We conclude with classes of posets for which we have conditions that are both necessary and sufficient.

31 pages, 20 figures. Incorporates correction to an ancillary theorem about caterpillar posets. Thanks to Doriann Albertin, Jean-Christophe Aval and Hugo Mlodecki for finding the error