Betti Numbers of Monomial Ideals and Shifted Skew Shapes

We present two new problems on lower bounds for Betti numbers of the minimal free resolution for monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are quadratic and bihomogeneous with respect to two variable sets, but gives a more finely graded lower bound. These problems are solved for certain classes of ideals that generalize (in two different directions) the edge ideals of threshold graphs and Ferrers graphs. In the process, we produce particularly simple cellular linear resolutions for strongly stable and squarefree strongly stable ideals generated in a fixed degree, and combinatorial interpretations for the Betti numbers of other classes of ideals, all of which are independent of the coefficient field. Partially supported by NSA grant H98230-07-1-0065 and by the Institute for Mathematics & its Applications at the University of Minnesota. Partially supported by NSF grant DMS-0601010. the electronic journal of combinatorics 16(2) (2009), #R3 1