Proving for which values an inequality holds in Mathematica

Question

I have the following inequality:

$\frac{(n-a)!}{(n-b)!} < \frac{(m-a)!}{(m-b)!}$,

where $n<m$ and $a,b,n,m\in \mathbb{N}$. I now need to derive sufficient conditions among $a$ and $b$ such that the inequality holds. (e.g. $a<b$). Can I do this with Mathematica and if so, how?

Answer 1

Perhaps in a while someone will remember a function which will give you the answer. But while we wait for them lets look at the answer.

z = Partition[
        Flatten[
            Table[
             {a, b, m, n, (n-a)! / (n-b)! < (m - a)! / (m - b)!},
             {a, 1, 4}, {b, 1, 4}, {m, 1, 4}, {n, 1, m-1}]],
    5];

zz = Cases[z, {_, _, _, _, True}]

which will throw a bunch of errors about dividing by zero for the cases that don't satisfy your inequality and show you only the results where your inequality is true, which gives you

{{1, 2, 2, 1, True}, {1, 2, 3, 1, True}, {1, 2, 3, 2, True}, {1, 2, 4, 1, True},
 {1, 2, 4, 2, True}, {1, 2, 4, 3, True}, {1, 3, 3, 1, True}, {1, 3, 3, 2, True}, 
 {1, 3, 4, 1, True}, {1, 3, 4, 2, True}, {1, 3, 4, 3, True}, {1, 4, 4, 1, True},
 {1, 4, 4, 2, True}, {1, 4, 4, 3, True}, {2, 3, 3, 2, True}, {2, 3, 4, 2, True},
 {2, 3, 4, 3, True}, {2, 4, 4, 2, True}, {2, 4, 4, 3, True}, {3, 4, 4, 3, True}}

Now we stare at that and look for relations between a,b,m,n that seem to yield True. In those results I see

Select[z, #[[1]] < #[[2]] && #[[2]] <= #[[3]] && #[[1]] <= #[[4]]&]

which gives exactly the same result as zz.

Thus a < b && b <= m && a <= n appears to be your necessary and sufficient conditions.

You can try removing any one of those relations and see that it fails. You can try substituting other relations and see if you can find anything simpler.

Test this carefully to make certain that I haven't made any mistakes.

And yes, people will be able to find a way to remove perhaps as many as a dozen characters from the code I've written, there are always a dozen different ways of writing anything in Mathematica. Pick one that you can remember and not make mistakes using.