I've been struggling with this simple problem for months though as I am a newbie to… well, maths, there's high chance someone more educated than myself may get it right!
Let's consider an array or a table or a Ferrers diagram or whatever it's called,
- $r$ rows by $c$ columns,
- in which each cell can either be blank or have a cross in it
- but each column must contain exactly $a$ crosses.
For instance, $\begin{array}{|c|c|c|c|c|c|} \hline ×&×&×&×& & \\ \hline ×&×& & &×&× \\ \hline & &×& &×& \\ \hline & & &×& &× \\ \hline \end{array}$ is such an array with $c=6,\ r=4,\ a=2$.
Now two arrays are said to be identical ('isomporphic' is probably the right word?) if we can obtain the second one from the first one by changing the order of columns and/or rows of the first one. So, the following array is identical to the one above:
$\begin{array}{|c|c|c|c|c|c|} \hline & & &×& &× \\ \hline ×&×& & &×&× \\ \hline &×&×& & & \\ \hline ×& &×&×&×& \\ \hline \end{array}$
but the order of columns is changed to $164325$ and the order of rows to $3241$.
Now the question is: how many unidentical (non-isomorphic) arrays are there, such that all the above conditions are satisfied and $c,r,a$ are given?
I know the answer for $c=6,r=4,a=2$ is $32$ because I have painstakingly went through every combination but how to do this more effectively?? :) I've been trying to learn enough group theory but it's all about rotating cubes or necklaces and I can't figure out how to translate this simple problem so that its tools can be deployed. Also, I tried to find a recurrence (by slicing away the last row and column) but it seems to be dependent on the actual placement of the crosses so…
I'd be glad for any help…