I won’t go into too much detail today, because it’s my first time back to the site in quite a while and I just want to get the information online. On the off-chance that this is a meaningful thing to think of, I want some kind of record that I figured at least the basics out.

As has probably been mentioned in previous posts, the inspiration for this line of inquiry came as early as late 2021, soon after I first learned of the crank of an integer partition as a concept. What was the issue? The properties of the crank are very nice, but the definition is incredibly ugly. I suspect that this has been “allowed” because most combinatorics happens in the realm of generating functions, and the generating functions surrounding the crank are perfectly fine-looking. So what did I want? I wanted a way to consider the crank, visually or just in relation to individual partitions, which made it actually seem like it might have meaning. In some sense, I wanted the explanation for why this ugly formula is special to be better than “if you work out the generating functions, they’re pretty nice.”

For context, the usual definition of the crank is as follows. For a partition P, let w(P) be the number of ones in the partition. If w(P) = 0, the crank is equal to the largest part of the partition. Otherwise, the crank is equal to the number of parts greater than w(P), minus w(P).

Very messy. I mean, it is impressive that it was even found. That said, I am not surprised that it was found primarily by working backwards from generating functions (at least, I suspect this, heaving read the paper that introduced the formula). It mixes the sizes of different parts with the numbers of other parts in multiple ways. It’s just messy. So here we start my new definitions. I apologize for the lack of LaTeX.

We define partitions of n to be equivalence classes of all permutations of n characterized by cyclic decomposition.

Definition 1: For a permutation R of n, a nonempty set of points X \subseteq {1, 2, …, n} is called cyclic if R(X) = X and \emptyset \subsetneq X’ \subsetneq X implies R(X’) /neq X’.

Definition 2: For a permutation R of n, a nonempty set of points X \subseteq {1, 2, …, n} is called fixed if R(X) = X and \emptyset \subsetneq X’ \subsetneq X implies R(X’) = X’.

In plain language, a cyclic set of points make up a single cycle, whereas a fixed set is made up exclusively of fixed points. These have been phrased in such a way to show that they are, in some sense, complementary concepts.

Definition 3: A partition P on n is canonical if, for any nonempty C \subseteq {1, 2, …, n}, (C is cyclic in some R in P) implies (C is fixed in no R in P).

Definition 3: A partition P on n is anti-canonical if, for any nonempty C \subseteq {1, 2, …, n}, (C is cyclic in some R in P) implies (C is fixed in some R in P).

These definitions are phrased to show their complementarity. In fact, one can show that for n >= 1, there are exactly as many canonical partitions of n as there are anti-canonical partitions of n.

However, if one wants a geometric understanding of these partitions, they can be phrased as: (a) a partition is canonical if it has no ones, and (b) a partition is anticanonical if no part is greater than the number of ones.