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### Honeycombs and Macrocyles

Small rings in chemistry are usually laid out as polygons; 5-membered rings as a pentagon, 6-membered as a hexagon, and so on. Once you get beyond about 9-10, this tends to look a little nasty. Or at least, unconventional.

So for 'macrocycles', it makes sense to make a less circular, and more wavy outline. Or, to be more exact, there is an inner cycle and several outer cycles, like this:

To make it clearer, I have used a chemical-like structure with oxygens in the inner ring. These crown ethers are a particularly clear-cut case, as the ethylene linkages force the particular geometry of the drawing. However, it is not so obvious for other sizes of rings - what possible arrangements are there?

Well, it occurred to me today that there is a simple formula for these macrocycle drawings. For a ring of size n with an inner ring of size i and outer rings or size o, you have to have n = (i * o) - i. The formula can be rearranged, but the idea is that you add up all the outer rings and remove the inner one.

So here are some crude representations of such cycles, for n in {9, 12, 14}. Below each cycle is the list of outer cycles. All of these examples are regular, in that they have the same number of vertices in the outer cycle. It is possible - although less desirable - to have different numbers of vertices in the outer cycles.

The version on the left is somewhat uglier than the 'puffed out' one on the left, but on some sense they are the same drawing. They are both (4, 6, 4, 6) in the notation of the previous image. Note that 24 = 4 + 6 + 4 + 6 - 4, which suggests that the previous formula can be generalised a bit to : n = (x0, ..., xr) - r. Each x in the equation is a ring size, and it is the sum of these minus the number of rings.

This immediately suggested a way to make examples - use partitions again! In other words, partitions of n + r for r in the range (3, floor(n / 3)). This produces a whole lot of horrible drawings, such as (3, 3, 9), but it does work. The code is here.

### How many isomers of C4H11N are there?

One of the most popular queries that lands people at this blog is about the isomers of C4H11N - which I suspect may be some kind of organic chemistry question on student homework. In any case, this post will describe how to find all members of a small space like this by hand rather than using software.

Firstly, lets connect all the hydrogens to the heavy atoms (C and N, in this case). For example:

Now eleven hydrogens can be distributed among these five heavy atoms in various ways. In fact this is the problem of partitioning a number into a list of other numbers which I've talked about before. These partitions and (possible) fragment lists are shown here:

One thing to notice is that all partitions have to have 5 parts - even if one of those parts is 0. That's not strictly a partition anymore, but never mind. The other important point is that some of the partitions lead to multiple fragment lists - [3, 3, 2, 2, 1] could have a CH+NH2 or an NH+CH2.

The final step is to connect u…

### 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) matrixA iff p*dominatesq 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 the column sum vector below.

Finally, the matrix can be interpreted as a bi…

### Havel-Hakimi Algorithm for Generating Graphs from Degree Sequences

A degree sequence is an ordered list of degrees for the vertices of a graph. For example, here are some graphs and their degree sequences:

Clearly, each graph has only one degree sequence, but the reverse is not true - one degree sequence can correspond to many graphs. Finally, an ordered sequence of numbers (d1 >= d2 >= ... >= dn > 0) may not be the degree sequence of a graph - in other words, it is not graphical.

The Havel-Hakimi (HH) theorem gives us a way to test a degree sequence to see if it is graphical or not. As a side-effect, a graph is produced that realises the sequence. Note that it only produces one graph, not all of them. It proceeds by attaching the first vertex of highest degree to the next set of high-degree vertices. If there are none left to attach to, it has either used up all the sequence to produce a graph, or the sequence was not graphical.

The image above shows the HH algorithm at work on the sequence [3, 3, 2, 2, 1, 1]. Unfortunately, this produce…