Question: From the wiki page [Catalan number](http://en.wikipedia.org/wiki/Catalan_number), we know the number of Young tableaux whose diagram is a 2-by-n rectangle given $2n$ distinct numbers is $C\_n$. In general, given $m\times n$ distinct numbers, how many Young tableaux whose diagram is a $m\times n$ rectangle are there? 

Also, what if these numbers can be repeated?

Many thanks.

Answer: 
For the answer to your main question, you need to use the [hook-length formula](http://en.wikipedia.org/wiki/Hook-length_formula#Dimension_of_a_representation).

[OEIS A060854](https://oeis.org/A060854) gives the result 
$$(mn)! \prod\_{i=0}^{n-1} \frac{i!}{(m+i)!} \textrm{ or equivalently } (mn)! \prod\_{j=0}^{m-1} \frac{j!}{(n+j)!} $$
and some more information.
It not quite clear what you mean by allowing repeated numbers, but what one usually considers in that case is so-called *semi-standard* Young tableaux, i.e., tableaux which are increasing (strict inequality) down each column, but only nondecreasing (equality allowed) along each row. The number of such arrangements on a given Young diagram, where the numbers $1,2,\dots,N$ are allowed, is counted as follows: define the "content" of box $(i,j)$ in the diagram to be $i-j$. Here's an illustration:

```
 0  1  2  3  4
-1  0  1  2
-2 -1  0
-3 -2
-4 -3
-5

```

Hook lengths are defined as for the usual hook-length formula for counting *standard* Young tableaux:

```
10  8  5  3  1
 8  6  3  1
 6  4  1
 4  2
 3  1
 1

```

To get the answer, take the product over all boxes of (($N$ plus the content of that box) divided by (the hook length for that box)).

(This is a special case of something called Stanley's hook-content formula.)