From propp@math.wisc.edu Sat Feb  9 10:37:14 2002
Date: Mon, 22 Oct 2001 20:22:55 -0500 (CDT)
From: propp@math.wisc.edu
To: reach@math.harvard.edu
Subject: [reach] minutes for 10/9

Minutes of October 9, 2001 (courtesy of Alex Healy)


Jim spoke about the purpose of weekly reports; in particular, report:

a) Progress
b) Hours spent
c) Plan for future work
d) How you feel about REACH
e) Suggestions for REACH (in response to d)

Computer infrastructure:  Who knows Matlab, Mathematica, Maple, MathCad, 
Magma.

Circulate AMM problem write-ups so that they're as good as possible by 
the deadline.
      Several have a solution to the Deutsch problem.
      Jim will bring in sample problems/solutions to "Monthly" problems 
on Thursday.
October 23rd meeting will be devoted (somewhat) to reviewing the 
"Monthly" write-ups (submit solutions to the group via e-mail on/by the 
18th).

Jim: How do we deal with the imbalance in (one-dimensional) 
combinatorial background? 
Break cliques and share ideas with everyone in the group.  More 
discussions, as a group.
More problems
More non-standard problems (like the negative Stirling recurrences, 
negative graph matchings).
Web kiosk
Clear write-ups of problems
Spelunking (i.e. "cave exploration"): feel free to explore tangentially
related, but interesting problems, in moderation.  Neat cave example
(courtesy of Jim):  merging Galois Theory and Combinatorics, e.g. the
Catalan number g.f. lives in a quadratic extension over the ring of 
formal power-series; what's the deal with its Galois conjugate?

Jim:  The main goal is still to work on reciprocity, backwards 
recurrences.

David Speyer will write up a guide to solving LREs this weekend.

Jim will setup an e-mail archive. 

Matt Lee will look into auto-archiving of the REACH e-mail list.

Topics of discussion for Thursday's meeting:
	Jim will say a bit more about combinatorics of sets with "negative"
		elements.
	Running Bow-tie recurrences backwards
	Present open problems early in the meeting
	Programming/hacking: algorithmic (numerical) solutions to problems

Roberto:  A real-world application of this combinatorics: Organic 
Chemistry

"Edges and grooves" by Jim:
Grooves = negative edges.  E.g. if you run an edge-by-edge 
graph-deleting algorithm backwards, you get "grooves" when you remove
edges that aren't there.
In the case of the "box"-graphs B_n (i.e. where #matchings = F_n), the 
number of matchings of these graphs is given by running the Fibonacci
recurrence backwards, where the sign is (-1)^{number of vertical edges}.
This is a possible direction for thinking about negative 
quantities/edges in combinatorics.

See /~propp/reciprocity.ps for some details of this sort of approach 
(specifically tilings of a 3-by-n or 4-by-n board)

It would be nice to extend this to a more general setting: how does this 
work when you tile a 3-by-n board with a "bite" taken out of the corner.
 What's a combinatorial interpretation for this?

Cylindrical tilings; i.e. mod out by some boundaries of a kxn rectangle. 
 (Moebius strip tilings?)

Domino tilings of n-by-n square (Jim, initiated by Lionel):
	(Henry Cohn's work:)
	Let A_n = Sqrt(# of domino tilings of the 2n-by2n square) x 2^{-n}
	A_n is always odd!  Also, A_{-n} (suitably interpreted) = +/- A_n
	A_n ~ c^{n^2}
	Work in 2-adic number to work the recurrence backwards.
	Is there a combinatorial interpretation for A_{-n}? (likely to be 
	very hard).  A reference:  "2-adic Behavior of Numbers of Domino 
	Tilings" (by Henry Cohn): (or search for "Henry Cohn" at 
	combinatorics.org)
Also Lior Pachter's work (search at combinatorics.org)

Jim: Ways to tile a hexagon with rhombi (lozenges).  These correspond to
perfect matchings of hexagonal graphs (honeycombs or carbon compounds). 
Analogies of Domino tilings with Lozenge tilings.  Run lozenge tilings
backwards.

Alex will provide a link to Richard Stanley's website w/ Polya notes
(from Math 192 last year).  Here's the link to the class notes: 
http://www-math.mit.edu/~rstan/192/notes.html   Section 7 has most of
the guts of "Polya theory".

Bowties:
	David ran some recurrences backwards, and they look "nice".

Run the general formula for the area/surface of an n-dimensional sphere
in reverse.