Max/Min Sums From Permutations

[Rainer Rosenthal]

----- Original Message ----- 
From: "Ralf Stephan" <ralf at ark.in-berlin.de>
To: <seqfan at ext.jussieu.fr>
Sent: Sunday, January 30, 2005 1:01 PM
Subject: Re: Max/Min Sums From Permutations


> > This test results in "True". Therefore at least as 
> > to the conjecture that the sequence for the minimums 
> > and A026035 being equal is correct for the first
> > one thousand terms. The coding is not elegant but 
> > it does work.
> 
> Maybe it helps for the proof of the conjecture that 
> we have for the sequence also the closed form
> 
> %F A026035 (1/12) [2n^3 + 4n - 3 + 3(-1)^n ].
> 

Dear Ralf and dear SeqFans,

here is a regular from de.sci.mathematik, who is
interested in the state of the exploration of these
nice sequences. His name is Wolfgang Thumser, whom
I cited some days ago (correction still pending):

==================================================
Wolfgang Thumser in de.sci.mathematik made
the suggestion for a correction:

Formula for A101986:    (x+9x^2+2x^3)/6

==================================================

In de.sci.mathematik he asked this evening, whether
he could join the discussion and if there was a
reference to the discussion. 
I would like to help him. This is what he wrote today:

> ich hab' so zum Spass 'mal die Permutationen mit maximaler und
> minimaler Produktsumme berechnet, daraus laesst sich das allgemeine
> Schema gewinnen (bis auf Spiegelung existiert jeweils genau eine
> Realisierung):

  +++ Translation: I have computed - just for fun - the
  +++ permutations with max and min product sums. This
  +++ should help in getting a general solution. There
  +++ is always only one solution (not counting mirrors).
> 
>  >>> summ(0)
> (0, [], 0, [])
>  >>> summ(1)
> (0, [1], 0, [1])
>  >>> summ(2)
> (2, [2, 1], 2, [2, 1])
>  >>> summ(3)
> (5, [3, 1, 2], 9, [2, 3, 1])
>  >>> summ(4)
> (12, [4, 1, 2, 3], 23, [2, 4, 3, 1])
>  >>> summ(5)
> (22, [5, 1, 3, 2, 4], 46, [2, 4, 5, 3, 1])
>  >>> summ(6)
> (38, [6, 1, 4, 3, 2, 5], 80, [2, 4, 6, 5, 3, 1])
>  >>> summ(7)
> (59, [7, 1, 5, 3, 4, 2, 6], 127, [2, 4, 6, 7, 5, 3, 1])
>  >>> summ(8)
> (88, [8, 1, 6, 3, 4, 5, 2, 7], 189, [2, 4, 6, 8, 7, 5, 3, 1])
>  >>> summ(9)
> (124, [9, 1, 7, 3, 5, 4, 6, 2, 8], 268, [2, 4, 6, 8, 9, 7, 5, 3, 1])
> 
> Weisst Du, wie weit die mit dem Beweis sind, d.h. kannst Du 'ne
> Referenz auf die Diskussionsrunde dort geben?

  +++ Translation: Do you know how far those SeqFans have
  +++ discussed this by now? Or can you please provide
  +++ a pointer to the ongoing discussion?

Kind regards,
Rainer Rosenthal
r.rosenthal at web.de