The Lagrangian Grassmannian is an important example in symplectic geometry, see [here][1] or [here][2] for details. It shares many similarities with the ordinary Grassmannians (as one would expect from the name). As is well-known, the cohomology ring of the Grassmannians has a very nice combinatorial description in terms of partitions, see this very nice M.O. [answer][3] for example. Does there exist an analogous description for the cohomology ring of the Lagrangian Grassmannians? 

More precisely:

(i)   How many generators does the ring have?
(ii)  What are its dimensions?
(iii) What is its multiplicative structure?


  [1]: https://ncatlab.org/nlab/show/Lagrangian+Grassmannian
  [2]: https://en.wikipedia.org/wiki/Lagrangian_Grassmannian#Maslov_index
  [3]: https://mathoverflow.net/questions/196546/hard-lefschetz-theorem-for-the-flag-manifolds