A003319

[Gottfried Helms]

Am 15.08.05 02:06 schrieb Paul D. Hanna:
> Hello Gottfried,
>         As is common with deep thinkers, you may be looking deeper into
> the question than is needed.

;-) quite right... Also thanks for this compliment...

> The matrix you gave is constructed by making all columns equal to the
> factorial sequence. 
> When all columns of a matrix are equal, certain operations applied upon
> that matrix such as matrix multiplication, inverse, logarithm, etc., will
> result in a matrix that likewise has all columns equal; 
> the g.f. of the column vector of the resultant matrix will equal the
> operation applied upon the g.f. of the column vector of the original
> matrix.  I believe that you are already aware of this fact.

Yes, that was rather obvious.

>   
> The appearance of A003319 arises due to the relation: 
> log(1 + x + 2*x^2 + 6*x^3 + 24*x^4 + 120*x^5 +...+ n!*x^n +...) =
> x + 3/2*x^2 + 13/3*x^3 + 71/4*x^4 + 461/5*x^5 + ... + A003319(n)/n*x^n
> +... (n>0)

(should have written down explicitely; thanks for that).
I fiddled with some simple modifications of the columns,
but with no specific goal, just for curiousness, but
with no specific result. So I think I leave this afternoon-
game.

>  
> If your question is directed more to the combinatorial interpretation of
> the above, 
> than it is beyond me. 

And me.... And that "permutations", as stated in the comment to
A003319, are in relatively simple connection to factorials
should not be so surprising - so I better stop wondering here ;-).

But that you also found that connection to partitioning
of powers into powers (of all the same base), which underlies
that same process may have some implications.

>  
> Regards,
> Paul
Thanks for the comment -

Gottfried Helms