This is the first post in a series about integer partitions (in which older posts will be continuously updated to mesh better with newer posts). My previous post on the topic had some issues, both mathematically and conceptually, and after realizing that I decided to start from scratch. The goal: figure out an alternate definition of the Crank invariant of a partition that feels motivated, unlike the original.

I think I’ve accomplished that, though I feel that my terminology could be better, since I accidentally wandered across a few fields of math I’ve never touched before, making up my own terminology along the way. Wherever possible, I will use what I believe to be the correct terminology.

A final note before actual content: don’t spoil me on where the math might go. I’m slowly trying to work out any further implications, and I’d much rather feel any accomplishment that will come with doing so by myself. Okay… actual content time.

We will be alternately be using two different sets of terminology when it comes to partitions. The first is considering partitions as finite non-increasing sequences of natural numbers, with each element in the sequence being called a part. The second is considering partitions as equivalence classes of permutations within symmetric groups, where the equivalence relation maintains cyclic decomposition. I will give a gentler introduction to each of these in a future post. For now, I just want to get to the first part of the point. I apologize in advance for the lack of LaTeX.

Definition 1: Given a permutation rho in S_n, a non-strict subset C of {1, 2, …, n} is called cyclic if: (1) rho(C) = C; and (2) for any non-empty, strict subset C’ of C, rho(C’) != C’.

Definition 2: Given a permutation rho in S_n, a non-strict subset C of {1, 2, …, n} is called fixed if: (1) rho(C) = C; and (2) for any non-empty, strict subset C’ of C, rho(C’) = C’.

Note first the fundamental relatedness of these two definitions. Note second some less opaque interpretations of these definitions. A cyclic subset of points in a permutation is simply the set of points that make up a single cycle in the cyclic decomposition. A fixed subset of points in a permutation is composed entirely of fixed points, or points that are, in themselves, an entire cycle.

Now for the important definitions. Again, don’t steal these, I’m still working on them.

Definition 4: A partition mu (on n points) is canonical if, for any subset C of {1, 2, …, n} such that C is cyclic for some permutation rho in mu, there exists no rho’ in mu for which C is fixed. That is, (C cyclic for some permutation in mu) => (C fixed for no permutation in mu).

Definition 5: A partition mu (on n points) is anti-canonical if, for any subset C of {1, 2, …, n} such that C is cyclic for some permutation rho in mu, there exists some rho’ in mu for which C is fixed. That is, (C cyclic for some permutation in mu) => (C fixed for some permutation in mu).

Again note first the symmetry between these two definitions. Again note second some simplified versions. A partition is canonical if every part is greater than the number of ones. A partition is anti-canonical if no part is greater than the number of ones. In fact, the canonical definition in particular we can make even simpler: a partition is canonical if it has no ones. I say this form last because it hides the connection to anti-canonical partitions.

That’s where I’ll end this post, though I do have more (and it involves actual results). In the meantime, I again apologize for both the lack of LaTeX as well as the lack of diagrams. I am quickly reaching the point where I may just pay up to allow them. Thanks for reading- and don’t spoil me!