Re Colin Mallows's questions

[N. J. A. Sloane]

Here is a follow-up from Colin:


Hi, Neil. 

    Sorry I missed A060854.   (It's embarrassing because I used to know 
about the hook formula!)   Also, I didn't express my second question 
very clearly.  I meant to ask for the number of ways an mxn matrix can be
filled with 1:mn (call the entries Z_ij) such that there exists a pair 
of real vectors x = (x1,...,xm) and y= (y1,...,yn) such that the numbers 
(xi+yj) are all distinct and appear in the same order  as the Z's, i.e.
so that xi+yj < xi'+yj' iff Z_ij < Z_i'j'.    I.e. again, the Z's
represent the order of the values {xi+yj}.     For small m,n all the
possible Z's can be realized in this way, but for larger m,n there are some
that cannot be.    For example, the array   1 3 4
                                            2 5 8
                                            6 7 9  cannot be.  

   Colin Mallows (colinm at research.avayalabs.com)


NJAS