Configuration number I do research in thermo-statistic and I have a maths problem i'dd like to solve.
Let's consider a checkerboard of R rows and C columns.
We want to know the number of available configurations to fill the board with N balls with the following rules :
 - we can put only one ball per square
 - there must be n_i balls on the ith column
 - there must be m balls on each row (m is the same for all rows)
So we'dd like to find out this number of configurations as a function of R, C, N, {n_i}, m.
 A: See the formula on page 399 in:
http://stat.gamma.rug.nl/Snijders_Psychometrika1991_Enumeration_Simulation.pdf
EDIT For a more general setup see:
MR2600999 (2011e:05005) 
Barvinok, Alexander(1-MI)
On the number of matrices and a random matrix with prescribed row and column sums and 0–1 entries. (English summary) 
Adv. Math. 224 (2010), no. 1, 316–339.
A: The case of $R~$ bounded and $C~$ large is not covered by the formulas of Barvinok. The only published case as far as I know is for the row and column sums both being uniform. See E. Rodney Canfield and Brendan D. McKay, Asymptotic Enumeration of Dense 0-1 Matrices with Equal Row Sums and Equal Column Sums, Electron. J. Combin., 12 (2005) R29, Theorem 4.
http://www.combinatorics.org/Volume_12/Abstracts/v12i1r29.html .
To do the case of more general column sums asymptotically, you just need to apply a suitable central limit theorem.  Each column corresponds to an $R$-dimensional random variable taking values in $\{0,1\}^R$. All these variables are independent, and the row sums are just the sums of these variables.  You need a local limit theorem in the lattice case, for example around Corollary 22.3 of R. N. Bhattacharya and R. R. Rao, Normal Approximation and Asymptotic Expansions,
John Wiley & Sons (NY, 1976).
Don't expect any exact answers except for very small $n_i$. You can compute some exact values by making a recurrence based on adding one more column, and also by other means (see Section 6 of my paper).
