Piecewise lexsegment ideals in exterior algebras

The problem of describing the Hilbert functions of homogeneous ideals of an exterior algebra over a field containing a fixed monomial ideal $I$ is considered. For this purpose the notion of a piecewise lexsegment ideal in an exterior algebra is introduced generalizing the notion of a lexsegment ideal. It is proved that if $I$ is a piecewise lexsegment ideal, then it is possible to describe the Hilbert functions of the homogeneous ideals containing $I$ in a way similar to that suggested by Kruskal and Katona for the situation $I=0$. Moreover, a generalization of the extremal properties of lexsegment ideals is obtained (the inequality for the Betti numbers).