Symmetry breaking in the positive integers

Young's lattice is uniquely associated with a single conjugation automorphism: \[ c : \mathbb{Y} \to \mathbb{Y} \] Likewise each positive integer is associated to an additive partition in Young's lattice by its prime signature by the monotone $s$ mapping. \[ s : \mathbb{Z}_+ \to \mathbb{Y} \] This associates a young diagram to each positive integer, so that we can relate the symmetries of Young's lattice to the asymptotic distribution of the prime numbers of the positive integers. In particular, we have in Young's lattice that square free numbers {1,1,1,...} and prime powers {n} are conjugate.

Proposition. let $\mathbb{Y}$ be Young's lattice and let $\{1,1,1,...\}$ a sum of ones then its conjugate is a singleton $\{n\}$.

Proof. let $\{1,1,1,...\}$ be a sum of ones partition. Then the conjugate partition is the set $\{u_1,u_2,...\}$ with $u_1$ the number of terms less then or equal to one, $u_2$ the terms less then or equal to two, etc. But since all terms are less then one this always terminates to $\{n\}$ which is a singleton term. As the conjugation function is an involution, it follows that the conjugate of $\{n\}$ is a partition of the form $\{1,1,1,...\}$. $\square$

What this implies for the positive integers is that prime powers and square-free numbers should be basically symmetrical. But if we look at the Reimann zeta determined distribution of the prime numbers that is not what we get. \[ \zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} \] The probability that a positive integer is square free is equal to $\frac{1}{\zeta(2)}$ which is about $60$ percent. In fact a majority, or over half, of all positive integers are square free. \[ \frac{1}{\zeta(2)} = \frac{6}{\pi^2} \approx 0.6 \] Let us compare this to the probability that a given positive integer is a prime power. This is determined by the prime nmuber theorem. \[ \lim_{x \to \infty} \frac{1}{ln(x)} \approx 0 \] The situation isn't really any better when we extended this to arbitrary powers of $\ln(x)$ to get the probabilities of prime squares, prime cubes, etc. So in short, the probability any number is a prime power is zero. \[ \lim_{x \to \infty} \sum_{n = 1}^{\infty} \frac{1}{ln(x)^n} \approx 0 \] We can now compare the probabilities that a positive integer are square free or a prime power using the results determined by the Reimann zeta function and the prime number theorem. \[ \text{square free} \approx 60\% \] \[ \text{prime power} \approx 0\% \] This is a startling result because you would expect by Young's lattice that square free numbers and prime powers should be symmetric, but the positive integers break this symmetry. There is a clear diversity bias among the positive integers, so that positive integers prefer a variety of different prime number factors over having a few that are the same.