Work log for REACH

Fall 2002

09.24 1:00
Attended REACH meeting.

09.25 0:30
Solved homework problem from first REACH meeting.

09.26 1:00
Attended REACH meeting.

Total for September: 2.5 hours
Cumulative total through September: 2.5 hours

10.01 1:00
Attended REACH meeting.

10.01 1:00
Started working on REACH web page: tried to reproduce "security through obscurity" technique under AFS.

10.02 0:30
Read Propp's domino-tiling reciprocity paper.

10.03 1:00
Attended REACH meeting.

10.08 2:00
Worked on homework from last REACH meeting: finding a combinatorial model for "partial matchings of a 2 x -n rectangle". Found a possible interpretation, but it wasn't very nice.

10.10 2:00
Attended REACH meeting.

10.13 0:30
Read "one-pagers" and started thinking about possible research projects. Currently the Markov numbers and two-dimensional recurrences sound interesting.

10.15 1:00
Created this web page.

10.17 1:00
Attended REACH meeting.

10.20 0:30
Started reading The Laurent phenomenon by Fomin and Zelevinsky.

10.21 0:30
Read more details of FZ.

10.21 2:00
Worked on the "frieze pattern" recurrence, with Mathematica. See the first section of my notes on two-dimensional recurrences.

10.22 0:30
Continued work on two-dimensional recurrences.

10.23 1:00
Generated list of monomials corresponding to the "number fence" recurrence.

10.24 0:30
Read Kuo's graphical condensation article.

10.25 1:00
Found a combinatorial interpretation of the sequence 1, 1, 2, 5, 14, 42, 131, 417, ... defined by a 3x3 determinant formula in terms of lattice paths in a diagonal strip of width 6. Want to apply to other initial conditions, in particular 1, 1, 2, 6.

10.25 1:00
Found the corresponding combinatorial interpretation of the sequence 1, 1, 2, 6, 21, 77, ..., in terms of lattice paths in another diagonal graph.

10.26 2:00
Wrote up an essay on this interpretation and related facts about the sequence. Learned how to use latex2html and did some background reading.

10.27 1:30
Found a combinatorial interpretation for the number fence recurrence. I'll post the details when I have time (most likely Friday).

10.29 1:00
Attended REACH meeting.

Total for October: 21.5 hours
Cumulative total through October: 24 hours

11.05 1:00
Attended REACH meeting.

11.06 1:30
Looked for height functions on groves.

11.07 1:00
Attended REACH meeting.

11.07 3:00
Talked about grove height functions with Gabriel and Inna after meeting.

11.08 1:00
Tried to find generating functions for groves on standard initial conditions analogous to those for Aztec diamonds.

11.09 1:00
Read Gabriel's cube recurrence article and David Speyer's crosses-and-wrenches article.

11.10 1:00
Still looking for algebraic height functions for groves. Discovered that the natural analogue to the Aztec diamond case does not give Laurent polynomials in the height variable (except in simple cases).

11.12 1:00
Attended REACH meeting. Noticed that the paths counted by the b_n sequence are the same paths used in enumerating tilings of an m x n rectangle for m = 3. Maybe this generalizes to larger m and larger determinants?

11.14 1:00
Attended REACH meeting.

11.19 1:00
Attended REACH meeting.

11.22 1:00
Thought some more about the relationship between the number fence recurrence and paths in a triangular grid.

Total for November: 13.5 hours
Cumulative total through November: 37.5 hours

02.04 2:00
Attended REACH meeting.

02.06 2:00
Attended REACH meeting.

02.08 2:30
Experimented with the sequence x, y, (y+1)/x, (((y+1)/x)^4+1)/y, ...; found the linear recurrences satisfied by the two alternating subsequences. Did the same for (a, b) = (2, 2) and tried to relate to the combinatorial interpretation with some success.

02.11 2:00
Attended REACH meeting.

02.13 2:00
Attended REACH meeting.

02.14 1:00
Worked out the analogue of cube-popping for the number frieze recurrence ("square-popping").

02.17 1:30
Read Jim's document on possible research projects. Realized that number frieze patterns are a special case of the polygon triangulation problem, and that this gives us a way to add more variables to the number frieze recurrence preserving the Laurent property (like edge variables in the cube recurrence). Found a combinatorial interpretation for these variables ("cut edges") and proved it with square-popping. Hopefully we can translate this back into the language of polygon triangulation.

02.18 1:00
More work on correspondence between number friezes and polygon triangulations.

Total cumulative time: 51.5 hours