A minor note to referees October 18, 2010

I frequently receive referee reports which reference particular lines of my document as “Page 13, line 20 …” Of course, I appreciate all of the careful reading. But this strikes me as a very useless way to refer to a location in my file.

It is extremely unlikely that, after responding to the referees other more general comments, page 13 will still have the same line breaks it did when I submitted it. I do have the original submitted file (advice to young authors — save a copy of the original submission!), so I can figure out where it was, but I then have to locate where the corresponding line is in my modified file and find the corresponding line to edit. It would be a lot more convenient for me if referees gave me a few words from the line they are referring to, so I could directly search the file for them. I try to do this; if you are a referee, I encourage you to do the same.

A peculiar numerical coincidence October 6, 2010

One of the questions I put on a recent take-home exam is to determine a generating function for the number of -vertex trees where the children of each vertex occur a specific order, and there is no vertex with precisely one child. For example, there are such trees on vertices.

.

The first few values of this sequence are

1, 0, 1, 1, 3, 6, 15, 36, 91 …

When I started computing these, I noticed a strange pattern: they were all triangular numbers. And that’s not all; they were:
1
0
1
1
2+1
3+2+1
5+4+3+2+1
8+7+6+5+4+3+2+1
13+12+11+10+9+8+7+6+5+4+3+2+1

Fibonacci triangular numbers!

I recounted the next term several times, in different ways. I finally confirmed that the pattern fails, by the smallest bit: The next Fibonnaci triangular is 231, and the number of trees on 8 elements is 232. This is one of the most persuasive false patterns I’ve encountered.

Keeping going, the sequences separate from each other: the tree sequence continues 603, 1585, 4213, 11298, 30537 while the Fibonacci triangulars are 595, 1540, 4005, 10440. The difference between the sequences is 1, 8, 45, 208, 858, 3276 … These are suspiciously round numbers, though I don’t see a pattern in them yet.

Consider this a thread either to discuss the patterns above, or to discuss your own favorite false patterns.

A suggestion for mathematical English October 4, 2010

I constantly find myself writing phrases like “the number of pairs with , and “. And then recasting them, because of the rule of mathematical typesetting that you should not have two formulas separated by a mere comma.

I used to think that the typesetting rule was a useless nuisance, but I have been persuaded by experience that it is too hard on human eyes to see that the comma is not internal to the formula. Tonight, I realized there is another alternative. What if we all collectively decided that it was grammatical to write “the number of pairs with and and “? Not only would this solve the typesetting problem, but it would be helpful to be reminded of which logical connector we were using throughout the list, rather than just at the end.

Is there a downside to this construction which I am missing?