Restricted Weak Compositions, Labelled Partitions, and Trees

So in the last post about listing trees I outlined a slightly cumbersome method to list trees from degree sequences. Thinking about it a bit more, it would probably be far easier to just list all trees on some number of vertices and filter out by degree sequence. I talked a little about the WROM algorithm in this old post which is a constant time generator of 'free' (unlabelled) trees.

Anyway, that's boring so I was trying out the more complicated approach. It looks like generating a single tree from a degree sequence is as simple as the Havel-Hakimi method. Connect the largest degree (dn) to the dn next largest degrees. Also maintain a list of vertices that have already been connected to, and then at the next step connect only to those not already connected to. So, for [3, 3, 2, 1, 1, 1, 1] we get:

You might notice that trees a) and c) are isomorphic. Below the trees labelled by degree are the same trees labelled by DFS discovery order, and below that the 'layout' of the tree as described in the WROM paper.

As it happens, I tried to use restricted weak compositions and what I call 'labelled partitions' to do this efficiently but it doesn't work so well yet. It seems like this could all be done far easier using just successor functions...

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chalky

I wanted to show something that hints at the things that the new architecture can afford us: This is using a Java2D graphics Paint object to make it look like chalk...kindof. It's a very simplistic way of doing it by making a small image with a random number of white, gray, lightgray, and black pixels. edit: it doesn't look so good at small scales some tweaking of stroke widths and so on is essential.

The Gale-Ryser Theorem

This is a small aside. While reading a paper by Grüner, Laue, and Meringer on generation by homomorphism they mentioned the Gale-Ryser (GR) theorem. As it turns out, this is a nice small theorem closely related to the better known Erdős-Gallai (EG). So, GR says that given two partitions of an integer ( p and q) there exists a (0, 1) matrix A iff p* dominates q such that the row sum vector r(A) = p and the column sum vector c(A) = q . As with most mathematics, that's quite terse and full of terminology like 'dominates' : but it's relatively simple. Here is an example: The partitions p and q are at the top left, they both sum to 10. Next, p is transposed to get p* = [5, 4, 1] and this is compared to q at the bottom left. Since the sum at each point in the sequence is greater (or equal) for p* than q , the former dominates. One possible matrix is at the top left with the row sum vector to the right, and th...

1,2-dichlorocyclopropane and a spiran

As I am reading a book called "Symmetry in Chemistry" (H. H. Jaffé and M. Orchin) I thought I would try out a couple of examples that they use. One is 1,2-dichlorocylopropane : which is, apparently, dissymmetric because it has a symmetry element (a C2 axis) but is optically active. Incidentally, wedges can look horrible in small structures - this is why: The box around the hydrogen is shaded in grey, to show the effect of overlap. A possible fix might be to shorten the wedge, but sadly this would require working out the bounds of the text when calculating the wedge, which has to be done at render time. Oh well. Another interesting example is this 'spiran', which I can't find on ChEBI or ChemSpider: Image again courtesy of JChempaint . I guess the problem marker (the red line) on the N suggests that it is not a real compound? In any case, some simple code to determine potential chiral centres (using signatures) finds 2 in the cyclopropane structure, and 4 in the ...

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