Betti numbers of the Springer fibers over nilpotent elements in $$gl_n({\mathbb {C}})$$ of Jordan form $$(2^b,1^{a-b})$$

In this paper, we compute the generating function for the Betti numbers of the Springer fibers over nilpotent elements in $$gl_n({\mathbb {C}})$$ of Jordan form $$(2^b,1^{a-b})$$ , where $$a\ge b\ge 1$$ .