# Calculating or estimating a combinatorial multivariate sum

Dear all,

I'm currently looking at a problem in which the following combinatorial product emerges:

$c(m_1,\dots,m_\lambda;n_1,\dots,n_\lambda)=\frac{m_1 !}{(m_1-n_1)!}\frac{(m_1+m_2-n_1)!}{(m_1+m_2-n_1-n_2)!} \dots\frac{(m_1+\dots+m_\lambda-n_1-\dots-n_{\lambda-1})!} {(m_1+\dots+m_\lambda-n_1-\dots-n_\lambda)!}$

or

$c(m_1,\dots,m_\lambda;n_1,\dots,n_\lambda)=\prod_{i=1}^\lambda\frac{m_i+\sum_{j=1}^{i-1}(m_j-n_j)}{\sum_{j=1}^{i}(m_j-n_j)}$,

where $m_i,n_i$ are positive integers (or zero) and $i$ is the index $i=1,\dots,\lambda$. Also, I constrain the sums: i) $\sum_{i=1}^\lambda n_i=N$, and ii) $\sum_{i=1}^\lambda m_i=M$. Finally, $M\geq N$.

The interpretation of the product can be done by considering a process in which $m_i$ distinguishable items of something are "produced" at time step $i$, and $n_i$ of them get "consumed" at that time step. For a specific time step, $n_i$ could be greater than $m_i$ because not all products $\sum_{j=1}^{i-1} m_j$ have been consumed. $c$ represents the number of ways in which this production/consumption process can take place.

Questions: 1) Is there any literature out there that looks at this problem? 2) Can anybody think of ways of reducing $c$ further?

3) My ultimate goal is to calculate (or analytically estimate) the sum of $c(m_1,\dots,m_\lambda;n_1,\dots,n_\lambda)$ over all possible "histories" $\sum_{i=1}^\lambda n_i=N$.

Thanks

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Ed Wolf says (in a now-deleted answer): BTW, I have the exponential generating function for this, but since I'm looking for an answer to 3), I was hoping to find something more direct. –  Anton Geraschenko Jul 26 '12 at 5:14