Message from J Propp re Somos Seqs

[N. J. A. Sloane]

>>From propp at math.wisc.edu  Sun Apr 28 23:49:33 2002
>>Delivered-To: njas at research.att.com
>>From: James Propp <propp at math.wisc.edu>
>>Date: Sun, 28 Apr 2002 22:49:33 -0500 (CDT)
>>To: njas at research.att.com
>>Subject: proposed message to seq-fan
>>
>>Neil,
>>
>>I can't remember whether seq-fan is moderated or not, but in case
>>it isn't, I wanted to check with you first to make sure you think
>>the following is appropriate, before I submit it.
>>
>>I should mention to you (and if you think it's appropriate I can
>>mention in the message) that I don't make a profit on these shirts;
>>I just charge whatever the place that makes the shirts charges me.
>>
>>Thanks,
>>
>>Jim
>>
>>
>>From: Jim Propp (propp at math.wis.cedu)
>>Subject: Somos sequences: new theorem, new shirt
>>
>>[with apologies to recipients of multiple copies of this message]
>>
>>Combinatorial interpretations of the Somos-4 and Somos-5 sequences
>>(the integer sequences
>>
>>        1,1,1,1,2,3,7,23,59,314,...
>>
>>and
>>
>>        1,1,1,1,1,2,3,5,11,37,83,...
>>
>>given by the recurrences
>>
>>                a(n-1) a(n-3) + a(n-2)^2
>>        a(n) = --------------------------
>>                         a(n-4) 
>>
>>and
>>
>>                b(n-1) b(n-4) + b(n-2) b(n-3)
>>        b(n) = -------------------------------
>>                           b(n-5) 
>>
>>respectively) have now been found, thanks to the combined efforts of
>>Julian West, Mireille Bousquet-Melou, and the members of the REACH 
>>team: Daniel Abramson, Trevor Bass, Gabriel Carroll, Dennis Clark,
>>Seth Kleinerman, Lionel Levine, Roberto Martinez, Gregg Musiker,
>>David Speyer, Jason Burns, Bo-Yin Yang, and myself.  
>>
>>In particular, there exists a geometrically-defined sequence of bipartite 
>>planar graphs A(n) (resp. B(n)) such that a(n) (resp. b(n) is the number
>>of perfect matchings of A(n) (resp. B(n)).  The proof that these graphs
>>have the desired property uses Eric Kuo's method of graphical condensation.
>>
>>These combinatorial interpretations of the Somos-4 and Somos-5 sequences 
>>establish the positivity of the coefficients of certain polynomials and 
>>Laurent polynomials that are associated with the two Somos sequences.  
>>(The cluster algebra method of Fomin and Zelevinsky is a powerful tool for 
>>proving Laurentness, but it has not thus far been able to prove positivity 
>>theorems.)
>>
>>A "wearable preprint" is available; go to
>>	http://www.math.harvard.edu/~propp/reach/front.jpg
>>	(or http://www.math.harvard.edu/~propp/reach/front.bmp)
>>and
>>	http://www.math.harvard.edu/~propp/reach/back.jpg
>>	(or http://www.math.harvard.edu/~propp/reach/back.bmp)
>>to see what the first few Somos-4 graphs look like.  (Thanks to
>>Trevor Bass for creating the graphics, and to Gabriel Carroll
>>for his helpful input.)  A preliminary description of what the
>>tee-shirt depicts is also available:
>>	http://www.math.harvard.edu/~propp/reach/shirt.html
>>
>>To order the T-shirt (available in sizes S, M, L, X, and XX, and in
>>any color as long as it's black) for $13.50 per shirt plus postage, 
>>send me email; I'll be taking orders until May 3.  (I may take orders 
>>afterwards too, but to be sure of getting your shirt, you should
>>order before the coming weekend.)
>>
>>"REACH" is short for "Research Experiences in Algebraic Combinatorics
>>at Harvard"; the group began meeting in Fall 2001, and will continue
>>to operate through Spring 2003 (excluding summer and winter breaks),
>>using students from the Boston area as research assistants.  If you 
>>know of any mathematically talented high school or college students 
>>in the Boston area who might want to be involved with REACH during 
>>the coming academic year, please let me know.
>>
>>Jim
>>