Question using Mathematica for symbolic combinatorial equalities and inequalities.

• To: mathgroup at smc.vnet.net
• Subject: [mg49913] Question using Mathematica for symbolic combinatorial equalities and inequalities.
• From: wendemu <wendemu at ipe.et.uni-magdeburg.de>
• Date: Fri, 6 Aug 2004 03:09:28 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Dear all, I'd be grateful for some help.

I am using Mathematica 5.0 for Microsoft Windows (November 18,2003)
I am using Mathematica for symbolic combinatorial equalities and
inequalities.

My first question is:

FullSimplify[Sum[Binomial[n, k] * Binomial[n, r - k], {k, 0, n}]]
results in

Out[45]=
Gamma[1 + 2 n]/
( Gamma[1 + 2 n - r] Gamma[1 + r] )

However, it is well known that, for n and r being non-negative integers,
the above result is Binomial[2*n, r].

How do I make Mathematica give results not in terms of Gamma functions,
if the arguments are non-negative integers?

Mathematica CAN verify this result (but then I have to know it beforehand,
which I do usually do not), since

FullSimplify[Sum[Binomial[n, k] * Binomial[n, r - k], {k, 0, n}] -
Binomial[2*n, r] ]
gives

0

as required.

My second questions is:
How do I evaluate whether combinatorial inequalities are true or false?
E.g.
Sum[Binomial[n, k], {k, 0, r}] < 2^n -1
is true for r < n-1.

Are there Mathematica commands which will produce this result?

Thank you for your help,

Andreas

```

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