Dear Experts,

I'm a graduate student, dealing with group-theory.

In my current research, I used the bound "Alexander Gruber" wrote about in this post: See Here (Actually, I have just found out that the bound is known and already mentioned online).

I was able to give a detailed proof for this claim myself of course , but since it seems like this bound is well known, I want to add a reference .

So... does anyone know about a reference for the bound Alexander Gruber gives in the post above? I want to add it to the Bibliography section.

Thanks for your help!


For general Hall Algebra,

Ian MacDonald's Symmetric Functions and Hall Polynomials http://books.google.com.hk/books/about/Symmetric_Functions_and_Hall_Polynomials.html?id=srv90XiUbZoC

For general reference about the number of vector subspaces over a finite p-field,

I suggest that you can google the keyword "subspaces over a finite p-field", here're some papers and some related problems arise in StackExchange:


And a popular ref. is the Goldman's paper which may contain the bounds:

J. Goldman and G.-C. Rota, The number of subspaces of a vector space, in \Recent progress in combinatorics" (W. Tutte, Ed.), pp. 75{83. Aca- demic Press, San Diego, CA, 1969.


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