[seqfan] Re: Stirling numbers of first kind

[franktaw at netscape.net]

-----Original Message-----
>From: Joerg Arndt <arndt at jjj.de>
>
>In  http://www.research.att.com/~njas/sequences/A000254
>the offset appears to be wrong, it should be 1

It might be easier to just change the s(n,2) in the definition to 
s(n+1,2) instead.  The rest of the definition is correct with the 
current offset, and most of the comments and formulas are correct with 
the current offset.  "Fixing" it would be a lot of work.  If somebody 
wants to undertake that, I won't object, but I don't think it's worth 
the effort.  Note that you would also have to check the 82 other 
sequences that reference this one to see to it that they are using the 
correct offset (if they use one at all).  This being such an important 
sequence, there may be references to it with the current offset in the 
mathematical literature.

>s(n,k) := numof length-n permutations into k cycles
>("Stirling cycle numbers")
>
>...
>
>EGF (serlaplace((-log(1-x))^2)/2!)
>0 + 0*x +  x^2 + 3*x^3 + 11*x^4 + 50*x^5 + 274*x^6 + ...

The E.g.f., as given by Michael Somos, is entirely correct, and covers 
both the offset 0 and offset 1 cases.

>Top comment should be
>"number of permutations of n elements into 2 cycles"

I would instead say "Number of permutations of n[+1] elements with 
exactly 2 cycles".  But this isn't really critical.

>For:
>A000399 Stirling numbers of first kind s(n,3).
>  http://www.research.att.com/~njas/sequences/A000399
>EGF is
> (serlaplace((-log(1-x))^3)/3!)
>0+0*x+0*x^2+  x^3 + 6*x^4 + 35*x^5 + 225*x^6 + ...
>and not (as given)
> (serlaplace((-log(1-x))^3))
>6*x^3 + 36*x^4 + 210*x^5 + 1350*x^6 + ...

Agreed.

>Top comment should be
>"number of permutations of n elements into 3 cycles"
>
>A000454 Stirling numbers of first kind s(n,4).
>top comment should be
>"number of permutations of n elements into 4 cycles"

See above.

>Correct EGF to (-log(1-x))^4)/4!
>Correct example

The example is not wrong.  It might be clearer if written differently.

>A000482 Stirling numbers of first kind.
>  http://www.research.att.com/~njas/sequences/A000482
>Add to title "s(n,5)"
>EGF should be (-log(1-x))^5)/5!
>add (mutatis mutandis) top comment
>
>Similarly for
> http://www.research.att.com/~njas/sequences/A001233
> http://www.research.att.com/~njas/sequences/A001234

Franklin T. Adams-Watters