Iterated one-step Pieri formula

Statement
This page gives a corollary of the Pieri formula that is obtained as the iterated one-step version. It is a statement that relates the irreducible representations of $$S_m$$ and $$S_n$$ for $$m \le n$$, where $$S_n$$ is viewed as the symmetric group on $$\{1,2,3,\dots,n \}$$ and $$S_m$$ is the subgroup that fixes all the elements $$\{m + 1, m+2, \dots,n \}$$ and hence permutes $$\{1,2,3,\dots,m \}$$.

Version for general $$m$$ and $$n$$
Let $$\mu$$ be an unordered integer partition of $$m$$ and $$\nu$$ be an unordered integer partition of $$n$$. Then, the following numbers are equal:


 * 1) The multiplicity of the irreducible representation of $$S_m$$ corresponding to $$\mu$$ in the restriction to $$S_m$$ of the irreducible representation of $$S_n$$ corresponding to $$\nu$$.
 * 2) The multiplicity of the irreducible representation of $$S_n$$ corresponding to $$\nu$$ in the induced representation to $$S_n$$ from the representation of $$S_m$$ corresponding to $$\mu$$.
 * 3) The number of directed paths in the Young lattice from $$\mu$$ to $$\nu$$.
 * 4) The number of ways to number the skew Young diagram obtained by subtracting the Young diagram for $$\mu$$ from the Young diagram for $$\nu$$.

Note that (1) and (2) are equal by Frobenius reciprocity, so the real content is in the equality with (3) and (4).

Version for $$m = n - 1$$
Let $$\mu$$ be an unordered integer partition of $$n - 1$$ and $$\nu$$ be an unordered integer partition of $$n$$. Then, the following numbers are equal:


 * 1) The multiplicity of the irreducible representation of $$S_m$$ corresponding to $$\mu$$ in the restriction to $$S_m$$ of the irreducible representation of $$S_n$$ corresponding to $$\nu$$.
 * 2) The multiplicity of the irreducible representation of $$S_n$$ corresponding to $$\nu$$ in the induced representation to $$S_n$$ from the representation of $$S_m$$ corresponding to $$\mu$$.
 * 3) The number that is 1 if $$\nu$$ can be obtained by adding a box to $$\mu$$, and zero otherwise.

Version for $$m = 1$$
In this case, $$\mu$$ must be the trivial partition of $$1$$. As before, let $$\nu$$ be an unordered integer partition of $$n$$. The following numbers are equal:


 * 1) The degree of the irreducible representation corresponding to $$\nu$$.
 * 2) The multiplicity of $$\nu$$ in the regular representation of $$S_n$$.
 * 3) The number of Young tableaux of shape $$\nu$$.
 * 4) The number of directed paths from the trivial partition to $$\nu$$ in the Young lattice.