Minutes for 10/22 Jim is not here, Anna will be running the meeting, Ananda is note taker. Anna will take notes on Thursday instead. Inna got 14 copies of the DVD Jim is getting the original DVD Is the noise in the room a problem? No one seems to have a problem with the noise. Gabriel's talk part two: Cube recurrence f(i,j,k)=(f(i-1,j,k)f(i,j-1,k-1)+f(i,j-1,k)f(i-1,j,k-1)+f(i,j,k-1)f(i-1,j-1,k))/f(i-1,j-1,k-1) f(i,j,k)=sum(w(G)) is the sum of groves The polynomials have coefficients all equal to 1 so each grove has a unique monomial. Gale-Robinson sequences sequences of the form s(n)s(n-k)=s(n-a)s(n-k+a)+s(n-b)s(n-k+b)+s(n-c)s(n-k+c) where a+b+c=k. Each term of the sequence is a Laurent polynomial. Anna asked whether Gale-Robinson is strongly monic. Gabriel says that this is probably not true, but the coefficients haven't been studied. [In reading over these notes, Jim says that in all the cases he's studied, the only way to get strong monicity to occur was to lift the recurrence into three dimensions.] Gale-Robinson is a special case of the cube recurrence. Taking f(i,j,k)=s(ia+jb+kc+n) proves that Gale-Robinson sequences have the Laurent property, and give some idea of the combinatorial meaning of the sequences. What does combinatorial interpretation of cube recurrence tell us about Gale-Robinson sequences? Is there a cleaner combinatorial interpretation for Gale-Robinson sequences? Other open questions relating to groves: Kuo condensation is a method for counting matchings of graphs. Take two graphs superimpose them and then separate them into two different graph matchings. See Kuo's paper at www.cs.berkeley.edu/~ekuo/condensation.ps Kuo's method did not work for groves. Take i+j+k between -1 and 1. Take two groves which are order 3 apart. Superimpose the two graphs, and then break into two graphs which are order one apart. There are several shapes of decomposition, each of which is one term. We want to describe what objects we get when we superimpose two groves. What is the algorithm for decomposing? We would like to know if each grove has a unique type of decomposition. This requires some empirical research. Is there a way to figure out how many decompositions occur? Relation between groves and matchings With appropriate choice of edge weights, one term in the cube recurrence can be dropped, so the cube recurrence is written as f(i,j,k)=(f(i-1,j,k)f(i,j-1,k-1)+f(i,j-1,k)f(i-1,j,k-1))/f(i-1,j-1,k-1) This is the same as the octahedron recurrence under a linear change of variables. Look at the matchings of the graph on the cube recurrence. We would like to discover grove-like objects on the octahedron recurrence. For more on octahedron recurrence, see David Speyer's paper www.math.harvard.edu/~propp/reach/speyer/candw.pdf on how octahedron recurrence counts matchings of graphs. Jim talked on thursday about alternating sign matrices and Aztec diamonds. There is something similar on groves called alternating sign triangles. Extract the triangle from the cube recurrence by labelling every other vertex with its degree-2. Anna asked how much intial conditions affect the graph. Answer: They affect the graph, we use these (i+j+k between -1 and 1) because they are easy. What rules govern an alternating sign triangle? It would be interesting to develop a theory of alternating sign triangles similar to that for matrices. There is not a unique correspondence between groves and alternating sign triangles. There are many groves per triangle (just as there are many TOADs [= Tilings of Aztec Diamonds] per alternating sign matrix). How many groves per triangle is an open question. Anton: If we are looking for alternating sign triangle, is there a reason that they correspond to groves? could they correspond to the sum of two triangles. Gabriel: Its possible. Rui: Why do we leave out half the vertices when creating the alternating sign matrix? Gabriel: From looking at the case where i+j+k=1 and also looking at the case i+j+k=-1, which is a slightly smaller triangle. Anna: If you look at the alternating sign triangle from superimposed groves can you decompose the triangle? Gabriel:That is an interesting quesion. For more information on alternating sign matrices see, D.P. Robbins and H. Rumsey, Jr., "Determinants and alternating sign matrices" in _Advances in Mathematics_ 62 (1986), 169-184 Jim has the question of circles in groves in the previews. Get random groves of large size, the probability of the edges being straight in corners goes to one. The random part seems to make a circle. We only have random sampling for standard initial conditions. It would be interesting to see what would happen for new intial conditions. It would be interesting to study the correspondence between octahedron and cube recurrence. There is a code that works by randomly enlarging the grove. Somebody could write java to have a nice visual aspect to random grove algorithm, similar to the aztec diamond one. Read Gabriel's paper at http://www.fas.harvard.edu/~gcarroll/math/gtpreach/cubeart.pdf or http://www.fas.harvard.edu/~gcarroll/math/gtpreach/cubeart.ps and send him editorial comments. Now we are going to go around the room and state what we are interested in. Anna: Groves but she is not so in the circles in groves problem. interested in generalizing to 3-D groves. philosophically: what makes a combinatorial object interesting? A cool property, or it needs two cool properties. For example groves have recurrsion. could we generalize them to something bigger, like 3-d. want to find the essential structure of groves. Rob: a 3-d grove would be a bunch of tetrahedrons. Inna: try to using cube recurrence method on the octahedron case. Gabriel: see what others are interested in about groves. Also interested in 3-adic behavior Siddique: Gale-Robinson sequences and p-adic look at 9-adic behavior. has started to generate some data, look at his webpage. Trevor:p-adic, has been looking at old stuff on 2-adic behavior Elnatan: polyhex tilings as well as alternating sign matrices and alternating sign triangles John B: Combinatorics and reciprocity of not necesarily perfect graphs Amanda: p-adic and lots of other things John G: 2nx2n domino counting, p-adic, looked at polyhex stuff Andy: polyhexes Anton: height functions on trihexes, groves, Markov numbers or some similar algebraic structure David: Polyhex tilings Alice: polyhexes Kyle: monomer-dimer tilings and cylinder tilings Kezia: Gale-Robinson and p-adic and groves Nick: Everything, mobius strips and markov numbers Ananda: p-adic and monomer-dimer Rui: Markov numbers, looked at 3-d snakes and trihexes Dan Z: polyhex Ilya: groves Reid: 3-d freize patterns Rob: generalize last years result to non-linear and interested in reciprocity We should all put our interests on our webpages. If we split into groups, put other people's names on your website. Jim should rearrange the names on the group website after we have divided into smaller groups. Jim would like us to make sure our smaller groups are not too large. Anna says that we should probably stay around 2-3 people per subgroup.