Catalan Inequality

For discussing Olympiad Level Combinatorics problems
Nirjhor
Posts:136
Joined:Thu Aug 29, 2013 11:21 pm
Location:Varies.
Catalan Inequality

Unread post by Nirjhor » Sun Oct 05, 2014 2:28 pm

If \(p(n)\) denotes the number of integer partitions of \(n\in\mathbb{N}\), prove that
\[p(n) < \dfrac{3n}{n+2} C_n\]
where \(C_n\) is the \(n\)th Catalan number for all natural \(n\).
- What is the value of the contour integral around Western Europe?

- Zero.

- Why?

- Because all the poles are in Eastern Europe.


Revive the IMO marathon.

Nirjhor
Posts:136
Joined:Thu Aug 29, 2013 11:21 pm
Location:Varies.

Re: Catalan Inequality

Unread post by Nirjhor » Mon Oct 06, 2014 12:59 am

Hint
$C_n$ counts the number of monotonic lattice paths from $(0,0)$ to $(n,n)$ not crossing the line $y=x$.
- What is the value of the contour integral around Western Europe?

- Zero.

- Why?

- Because all the poles are in Eastern Europe.


Revive the IMO marathon.

Post Reply