[seqfan] Re: New or old seq: The maximum number of shift-minimal winning coalitions, "On Dedekind's problem for complete simple games"

[franktaw at netscape.net]

1) The comment in A025591, "If k is allowed to approach infinity, this 
gives the partition numbers A000009. " is not correct.  The maximum of 
this sequence comes (after the first few values) from a power of x 
greater than n.  It would be appropriate to have a cross reference to 
A000009.

2) It would really help if two more values were calculated for the 
game-theoretic sequence (one would be useful, but less good).  If two 
more values match, I would just add a comment "It appears that this is 
the same as ...", and include the reference.  If they don't match, of 
course, you have your answer.

Equally good, of course, would be a proof that the sequences are the 
same.  You might try emailing the authors and see if they can prove 
equality.  I'm sure they would love to be able to publish it.

Failing either of the above, I would add it as a new sequence, with 
appropriate cross-refs to and from A025591.  If even one more value can 
be found to match, I would probably favor just adding a comment (as 
above).

Neil might give different answers; this is not an authoritative 
response.

Franklin T. Adams-Watters

-----Original Message-----
From: Jonathan Post <jvospost3 at gmail.com>

1, 1, 2, 2, 3, 5, 8, 14, 23, 40, 70, 124, 221, 397, 722

Table 1. The maximum number of shift-minimal winning coalitions.

p.5 of http://arxiv.org/abs/1001.3045
On Dedekind's problem for complete simple games
Authors: Sascha Kurz, Nikolas Tautenhahn
Comments: 13 pages, 1 figure, 6 tables, submitted to ISSAC 2010
Subjects: Combinatorics (math.CO)

We combine the parametric Barvinok algorithm with a generation
algorithm for a finite list of suitably chosen discrete sub-cases on
the enumeration of complete simple games, i.e. a special subclass of
monotone Boolean functions. Recently, Freixas et al. have proven an
enumeration formula for complete simple games with two types of
voters. We will provide a shorter proof and an enumeration formula for
complete simple games with two shift-minimal winning coalitions.

I note that this (except for a-0) agrees with
A025591         Maximal coefficient of Product_{k<=n} (x^k+1). Number of
solutions to +- 1 +- 2 +- 3 +- ... +- n = 0 or 1.

So is this a new seq, or a comment on an old one?


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