The functionality outlined below, and much more, is implemented in Hodge diamond cutter, which can be used in Sage. If you use it for your research, please cite it using DOI.

Just a quick service announcement: I have now added Hodge diamonds for horospherical varieties, as discussed in Gonzales--Pech--Perrin--Samokhin, Geometry of horospherical varieties of Picard rank one. Maybe I should spend some time discussing their geometry, but this will be it for now, interested readers are referred to this excellent article.

This is part of an effort to add horospherical varieties to, even though they are not quite homogeneous varieties, their geometry is sufficiently close to warrant their inclusion in my opinion.

The usual comment applies: feel free to suggest other varieties whose Hodge diamonds you would like to be implemented!