The purpose of this post is mostly to point out that this is not exclusively a philosophy blog. That said, the site doesn’t currently have a LaTeX plugin, so the math is gonna be a bit janky to communicate. If you feel you need it, an introduction to the topic can be found at https://en.wikipedia.org/wiki/Partition_(number_theory).

The definition of a partition used in George Andrews’ “The Theory of Partitions” is “a finite nonincreasing sequence of positive integers.” This definition is perfectly fine and I intend to build on it here, not truly replace it. The motivation for my altered definition is the definition of the Crank invariant of a partition. Its standard definition is as follows: if the partition contains no ones, the Crank is equal to the largest element of the partition. Otherwise, it equals the number of elements larger than the number of ones in the partition.

I very much wanted to get rid of the conditional, which seemed to me to indicate something imperfect about the definition. Further, the Crank behaves similarly in many respects to another, simpler invariant called the Rank, which is defined as the largest element in the partition minus the number of parts in the partition. The considerable difference in the form of their definitions also bothered me. Note that my alternate Crank definition isn’t fundamentally different from the original; it provides a value equal to the standard Crank of the partition conjugate to the one it considers. As conjugation is well-defined and bijective on all partitions of a given positive integer, this switching maintains all of the partition-organizing properties of the original definition while being more aesthetically pleasing. Because of my lack of LaTeX, we will call N the standard natural numbers and N_0 the natural numbers including zero.

We redefine a partition to be any function p: N_0 -> N_0 such that it has the two following properties for 0 < j < k:
(1) p(j) >= p(k), and
(2) [p(j) = 0] if and only if [j > p(0)].

Note that if I had LaTeX, I would phrase it as “p(j) is not greater than 0 exactly when j is greater than p(0)” or something like that. The differences between this and the standard definition are twofold. First, the sequence is no longer finite, but instead features infinite trailing zeroes. Second, the sequence has a zeroth term equal to the number of parts of the partition.

A nice side effect of this definition is that the Zero Partition is well-defined in this scheme, letting the typical generating sequences for partition-counting make a little more sense. I still need to work on the phrasing a bit more, to make this even clearer.

The primary benefits, however, are how it allows us to redefine the Rank and Crank invariants into clear and clearly parallel expressions:
We redefine the the Rank of a partition p to be [p(0) – p(1)];
We redefine the Crank of a partition p to be [p(p(1) – p(2))].

I’m not particularly well-versed in the study of integer partitions beyond a basic level, so I don’t know if this convention is useful in other places, but I intend to check, so nobody spoil it for me. Thanks!