# Colin Mallows's problems

Roland Bacher Roland.Bacher at ujf-grenoble.fr
Fri Jun 10 17:37:58 CEST 2005

```If I understand correctly,

this is the number of standard tableaux of shape n x m.
(Fillings of an n x m grid with integers 1,... , nm,
strictly increasing in each row and column).
It corresponds to the dimension of an irreducible representation
of the symmetric group of nm elements and this dimension
is given (for instance) by the hook formula:
(nm)!/product over all hook lengths
where a hook is as follows

**********
****xxxxxx
****x*****
****x*****

Example 3 times 3:  9!/(5*4*3*4*3*2*3*2*1)=42

Did I get something wrong?  Roland Bacher

On Fri, Jun 10, 2005 at 10:59:44AM -0400, N. J. A. Sloane wrote:
> Mitch says:
>
> > >  For each m,n, how many of these arrangments can be realised as the ordering
> > >of numbers of the form x_i + y_j?
> >
> > Is there an example for this? I don't understand how this is supposed to
> > work.
>
>
> Me:  i assumed it was something like this:   certain of
> these arrangements can be obtained by taking m numbers x1 ... xm
> in increasing order of course
> and n numbers y1 ... yn, and filling the array with M_{i,j} = xi + yj.
>
> If we are lucky, this will use all the numbers 1 ... mn exactly once
> - in how many ways can this be done?
>
> NJAS

```