Reconstructing Partitions from their Multisets of $k$-Minors

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 Naruses 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 $lambda$.