Minutes for 2/12/04

Today's snacks: Stephen (next time: Martin)
Today's notes:  Carl   (next time: Sam)

HAL  with the recurrence a_n * a_{n-4} = a_{n-1} * a_{n-3} + 1
     (1,1,1,1,2,3,4,9,14,19...)  we showed how to count hexagonal
     matchings that correspond to:
                                     3
                                  4    14              
                                    19      (AB=CD+1)
     using Linstrom's Lemma.
     Now, we want to try to "superimpose" 3 & 19
     (that is, draw the hex-graph corresponding to 3 on top of 19)

     "So where do you put it?"  "The natural place."

the "3"  __                               A 
      __/  \    this is the 'A' in the  C   D  picture.
     /  \__/1                             B  
     \__/2   
      3         'C' adds one row to the bottom-left to 'A'
                'D' adds one row to the bottom-right to 'A'
  and 'B' adds both ('C' and 'D') to 'A'

                     
the "19"             __
                  __/1 \
        1__    __/1 \__/
     1__/  \__/1 \__/
  1__/  \__/1 \__/
  /  \__/2 \__/               
  \__/3         
   4


Now, to superimpose 3 & 19, me get a "multi-matching"
(different from a multi-graph). Now, we get to use each
repeated vertex twice.  (we get something like this: )
                      __
                   __/  \
                __/  \__/
        //=\\__/  \__/
   __//=\\=//  \__/
  /  \\=//  \__/
  \__/


What results is either loops or paths.  There might be something
interesting counted by the number of paths in these matchings..?

EMILIE put the recurrence a_n * a_{n-3} = a_{n-1} * a_{n-2}
      + a_{n-1} + a_{n-2}
 into 3D.  A=(CD+E+B)/F

           
from      F      from          F     from          F
west:    / \     southwest:    |     northwest:    |
        D   E                 D+E               D--+--E
        |   |                  |                   |
        B   C               B--+--C               B+C
         \ /                   |                   |
          A                    A                   A


        These recurrences seem to be Laurent, but not "faithful".
	[Actually, we learn later that they aren't even Laurent:
	see below.]

HAL/SAM  more talk with number walls.  Hal wants to look for a
         class of combinatorial objects that grows like the
         hexagonal graph matchings, in a way that is understandable.

JIM  goes over Emilie's 3D recurrence.  looks at it from another angle
     (C=(AF-B-E)/D).

     With  a_n * a_{n-3} = a_{n-1} * a_{n-2} + a_{n-1} + a_{n-2}, we
     might be able to find a number-theory proof that the sequence
     contains only integers, using techniques similar to those in
     the Somos article.

:: break ::

mmmm... bagels....  (Thanks Stephen!)

JIM  What about un-seeded bagels?  Should we avoid the unnecessary mess?

:: to Lab ::

B107 hums with machines of maple math, but few audible words are spoken.

EMILIE's 2D recurrence seems not to work.  Though, the 1D recurrence
         produces integers and is Laurent (at least for many many terms)
         Stuff she did today can be found in maple worksheet format at:
         <www.sit.wisc.edu/~eahogan/folder/darnsequence2.mws>

MARTIN  suggests Magma software is more efficient, can probably compute
        stuff farther than maple. (investigate getting a site license?)

SSL officially ends.  Sam and Carl test their newfound B107 privileges
    and stay after, UNATTENDED.  If only they had brought camping gear.