[proofplan]
We derive the vertical Pieri rule from the [horizontal Pieri rule](/theorems/5205) by applying the standard involution $\omega$ on the ring of symmetric functions. The involution sends complete symmetric functions to elementary symmetric functions and sends Schur functions to Schur functions indexed by conjugate partitions. Conjugation of Young diagrams converts horizontal strips into vertical strips, so the indexing set in the horizontal Pieri formula transforms exactly into the indexing set in the desired formula.
[/proofplan]
[step:Apply the symmetric-function involution to the horizontal Pieri rule]
Let $\Lambda$ denote the ring of symmetric functions over $\mathbb{Z}$. For every integer $k \geq 0$, let $h_k \in \Lambda$ denote the complete symmetric function of degree $k$, and let $e_k \in \Lambda$ denote the elementary symmetric function of degree $k$. Let $\omega: \Lambda \to \Lambda$ denote the standard involutive ring automorphism characterized by
\begin{align*}
\omega(h_k) = e_k
\end{align*}
for every integer $k \geq 0$. This involution satisfies
\begin{align*}
\omega(s_\nu) = s_{\nu'}
\end{align*}
for every partition $\nu$, where $\nu'$ denotes the conjugate partition. This is the standard Schur-function involution identity, obtained by applying $\omega$ to the Jacobi-Trudi determinant for $s_\nu$ and using the conjugate Jacobi-Trudi determinant for $s_{\nu'}$.
Apply the Horizontal Pieri Rule to the partition $\mu'$ and the integer $r$. It gives
\begin{align*}
h_r s_{\mu'} = \sum_{\nu} s_\nu,
\end{align*}
where the sum ranges over all partitions $\nu$ such that $\nu / \mu'$ is a horizontal strip of size $r$, meaning that the skew diagram $\nu / \mu'$ has $r$ boxes and contains at most one box in each column.
Applying the ring homomorphism $\omega$ to both sides gives
\begin{align*}
\omega(h_r s_{\mu'}) = \omega\left(\sum_{\nu} s_\nu\right).
\end{align*}
Since $\omega$ is a ring homomorphism and is additive, this becomes
\begin{align*}
\omega(h_r)\omega(s_{\mu'}) = \sum_{\nu} \omega(s_\nu).
\end{align*}
Using $\omega(h_r)=e_r$, $\omega(s_{\mu'})=s_{(\mu')'}=s_\mu$, and $\omega(s_\nu)=s_{\nu'}$, we obtain
\begin{align*}
e_r s_\mu = \sum_{\nu} s_{\nu'},
\end{align*}
where $\nu$ ranges over all partitions for which $\nu / \mu'$ is a horizontal strip of size $r$.
[guided]
The purpose of introducing $\omega$ is to convert the complete symmetric function $h_r$ in the horizontal Pieri rule into the elementary symmetric function $e_r$ appearing in the vertical Pieri rule.
Let $\Lambda$ denote the ring of symmetric functions over $\mathbb{Z}$. For every integer $k \geq 0$, let $h_k \in \Lambda$ denote the complete symmetric function of degree $k$, and let $e_k \in \Lambda$ denote the elementary symmetric function of degree $k$. Let
\begin{align*}
\omega: \Lambda \to \Lambda
\end{align*}
be the standard involutive ring automorphism satisfying
\begin{align*}
\omega(h_k) = e_k
\end{align*}
for every integer $k \geq 0$. The same involution acts on Schur functions by conjugating the indexing partition:
\begin{align*}
\omega(s_\nu) = s_{\nu'}
\end{align*}
for every partition $\nu$. This follows from the Jacobi-Trudi formula for $s_\nu$, because applying $\omega$ replaces complete symmetric functions by elementary symmetric functions, and the conjugate Jacobi-Trudi formula identifies the resulting determinant with $s_{\nu'}$.
We now apply the Horizontal Pieri Rule, not to $\mu$, but to the conjugate partition $\mu'$. This choice is forced by the identity $\omega(s_{\mu'}) = s_\mu$. The horizontal Pieri rule gives
\begin{align*}
h_r s_{\mu'} = \sum_{\nu} s_\nu,
\end{align*}
where the sum is over all partitions $\nu$ such that $\nu / \mu'$ is a horizontal strip of size $r$. Here horizontal strip means that the skew diagram $\nu / \mu'$ has $r$ boxes and contains at most one box in each column.
Apply $\omega$ to both sides. Since $\omega$ is a ring homomorphism, it preserves products, and since it is additive, it preserves finite sums. Therefore
\begin{align*}
\omega(h_r s_{\mu'}) = \omega\left(\sum_{\nu} s_\nu\right)
\end{align*}
becomes
\begin{align*}
\omega(h_r)\omega(s_{\mu'}) = \sum_{\nu} \omega(s_\nu).
\end{align*}
Now substitute the defining identities for $\omega$:
\begin{align*}
\omega(h_r)\omega(s_{\mu'}) = e_r s_{(\mu')'} = e_r s_\mu,
\end{align*}
and
\begin{align*}
\omega(s_\nu) = s_{\nu'}.
\end{align*}
Thus
\begin{align*}
e_r s_\mu = \sum_{\nu} s_{\nu'},
\end{align*}
where the indexing condition is still that $\nu / \mu'$ is a horizontal strip of size $r$.
Now define $\lambda := \nu'$ for each partition $\nu$ in the sum. Conjugation is an involution on partitions, so this relabeling is bijective and $\nu = \lambda'$. The containment $\mu' \subset \nu$ is equivalent, after conjugating Young diagrams, to $\mu \subset \nu' = \lambda$. Conjugation preserves the number of boxes, hence
\begin{align*}
|\lambda / \mu| = |\nu / \mu'| = r.
\end{align*}
Finally, conjugation interchanges rows and columns. Therefore the condition that $\nu / \mu'$ has at most one box in each column is equivalent to the condition that $\lambda / \mu$ has at most one box in each row. This says exactly that $\lambda / \mu$ is a vertical strip of size $r$.
Relabeling the sum by $\lambda = \nu'$ gives
\begin{align*}
e_r s_\mu = \sum_{\lambda} s_\lambda,
\end{align*}
where the sum ranges over all partitions $\lambda$ such that $\lambda / \mu$ is a vertical strip of size $r$. This is the desired formula.
[/guided]
[/step]
[step:Relabel conjugate partitions and convert horizontal strips into vertical strips]
For every partition $\nu$ appearing in the last sum, define
\begin{align*}
\lambda := \nu'.
\end{align*}
Then $\nu = \lambda'$, and conjugation of partitions is a bijection from partitions $\nu$ with $\nu / \mu'$ a horizontal strip of size $r$ to partitions $\lambda$ with $\lambda / \mu$ a vertical strip of size $r$.
Indeed, the containment $\mu' \subset \nu$ is equivalent, after conjugating diagrams, to $\mu \subset \nu' = \lambda$. Conjugation preserves the number of boxes in the skew diagram, so
\begin{align*}
|\lambda / \mu| = |\nu / \mu'| = r.
\end{align*}
Finally, conjugation interchanges rows and columns. Hence the condition that $\nu / \mu'$ contains at most one box in each column is equivalent to the condition that $\lambda / \mu$ contains at most one box in each row. Thus $\lambda / \mu$ is a vertical strip of size $r$.
Relabeling the sum by $\lambda = \nu'$ therefore gives
\begin{align*}
e_r s_\mu = \sum_{\lambda} s_\lambda,
\end{align*}
where the sum ranges over all partitions $\lambda$ such that $\lambda / \mu$ is a vertical strip of size $r$. This is the desired formula.
[/step]
