Denote with $\mathcal{M}$ the set of $(m \times n)$matrices with integer coefficients bounded by some $K$. Given a matrix $B \in \mathcal{M}$ that is in Smith normal form, is anything known about the number of matrices $A \in \mathcal{M}$ that have Smith normal form $B$?

3$\begingroup$ this is essentially about counting presentations of a f.g. Abelian group with a given upper bound on size. $\endgroup$ – Dima Pasechnik Apr 30 '15 at 8:43

$\begingroup$ What do you want to know? There are obvious bounds... $\endgroup$ – Igor Rivin Apr 30 '15 at 20:52

4$\begingroup$ Yinghui Wang has just written a paper on this topic. Soon it should be posted on the arXiv. $\endgroup$ – Richard Stanley Apr 30 '15 at 23:52

$\begingroup$ thanks for your comments. @Igor I want to know as much as possible ;) The exact number if possible, if not then some "good" bounds. $\endgroup$ – Martin May 6 '15 at 13:24

8$\begingroup$ @Martin The paper is now available at arxiv.org/abs/1506.00160. I have been added as a coauthor. $\endgroup$ – Richard Stanley Jun 2 '15 at 1:29
Just to comment: another view of the question concerns representing elements of $\mathcal{M}$ in the form $PBQ$, for $P$ and $Q$ invertible integer matrices; in particular how big one need to take entries of $P$ and $Q$ to make sure that every element of $\mathcal{M}$ with the normal form $B$ is obtained as $PBQ$.