A question about Fomin's local rules for growth diagrams

Question

Let $w\in S_n$. Define the growth diagram of $w$ as follows: Start by an array of $n\times n$ squares, with an $X$ in the i'th column and row $w(i)$ from bottom. Then we obtain $(n+1)^2$ vertices (the corners of these squares). Label each vertex in the bottom row and the left-most column with the partition $\emptyset$ and use the following "local" rules to label the remaining vertices of the array: First, assume that given a square $s$, the bottom-left vertex is labelled with the partition $\lambda$, the bottom-right vertex is labelled with $\nu$ and the top-left vertex is labelled with $\mu$. We are going to label the top-right vertex with a partition $\rho$ as follows:

1. If $s$ doesn't contain a $X$ and if $\lambda=\mu=\nu$, set $\rho=\lambda$.
2. If $s$ doesn't contain a $X$ and if $\mu\neq \nu$, set $\rho = \mu \cup \nu$, that is $$ \rho_i = \max(\mu_i,\nu_i). $$
3. If $s$ doesn't contain a $X$ and if $\mu = \nu \neq \lambda$, then this implies that the skew diagram $\mu\setminus\lambda$ consists of a single box. Assume this box is in row $i$ of $\mu$. Then define $\rho$ by adding a box to row $i+1$ of $\mu$.
4. If $s$ contains a $X$, this implies that $\mu = \nu = \lambda$, and we define $\rho$ by adding a box to the first row of $\lambda$.

These are known as the Fomin's local rules. Now, I am having troubles understanding these rules. To be precise:

• I don't understand why, in rule 3, given that $\mu = \nu \neq \lambda$, this implies that $|\mu\setminus\lambda| = 1$.
• Similarly, I don't know why, in rule 4, if there is an $X$ in the square $s$ this implies that $\lambda=\mu=\nu$.

For a reference with an example, see Stanley & Fomin, Enumerative Combinatorics, Vol. II, Section 7.13.

I've worked several examples to try to figure it out, but I can't find a convincing explanation of this phenomena.

Another interesting property is the following: If a vertex $p$ is labelled with the partition $\lambda$, then $|\lambda| = \lambda_1 + \lambda_2 + \cdots$ is equal to the number of $X$'s appearing to left and below this vertex. I can't find a convincing explanation of this either.

I'm interested in growth diagrams, because they give a nice and symple proof of the symmetry of the RSK algorithm. I know other proofs, for example a proof involving antichains in the inversion poset of the two lines array of $w$, or the one using the matrix-ball construction of Viennot and Fulton, but I find the proof using the growth diagram very enlightening, once I have completely understood why these rules work.

Thank you in advance!