The cd-index of a semi-Eulerian poset

The numbers of flags of different ranks of an Eulerian poset are subject to a set of linear relations which have been described by Bayer and Billera. Fine showed that these relations are equivalent to the existence of a certain polynomial in two non-commutative variables, called the cd-index. The coefficients of such polynomial have been proved to be nonnegative for specific families of posets, hence revealing inequalities among the numbers of flags of such objects. In our work we show that it is possible to extend the definition of the cd-index from Eulerian to semi-Eulerian posets by a small modification of the flag f-polynomial. When the poset is simplicial, we show that a formula due to Stanley continues to hold in the semi-Eulerian setting. Our main result shows that the coefficients of the cd-index of a semi-Eulerian Buchsbaum simplicial poset are nonnegative, hence confirming a conjecture of Novik. This is joint work with Martina Juhnke and José Samper.