  For non-negative integers $n$ and $k$ with $n \ge k$, a {\em $k$-minor} of a
partition $\lambda = [\lambda_1, \lambda_2, \dots]$ of $n$ is a partition $\mu
= [\mu_1, \mu_2, \dots]$ of $n-k$ such that $\mu_i \le \lambda_i$ for all $i$.
The multiset $\widehat{M}_k(\lambda)$ of $k$-minors of $\lambda$ is defined as
the multiset of $k$-minors $\mu$ with multiplicity of $\mu$ equal to the number
of standard Young tableaux of skew shape $\lambda / \mu$. We show that there
exists a function $G(n)$ such that the partitions of $n$ can be reconstructed
from their multisets of $k$-minors if and only if $k \le G(n)$. Furthermore, we
prove that $\lim_{n \rightarrow \infty} G(n)/n = 1$ with $n-G(n) = O(n/\log
n)$. As a direct consequence of this result, the irreducible representations of
the symmetric group $S_n$ can be reconstructed from their restrictions to
$S_{n-k}$ if and only if $k \le G(n)$ for the same function $G(n)$. For a minor
$\mu$ of the partition $\lambda$, we study the excitation factor $E_\mu
(\lambda)$, which appears as a crucial part in Naruse's Skew-Shape Hook Length
Formula. We observe that certain excitation factors of $\lambda$ can be
expressed as a $\mathbb{Q}[k]$-linear combination of the elementary symmetric
polynomials of the hook lengths in the first row of $\lambda$ where $k =
\lambda_1$ is the number of cells in the first row of $\lambda$.
