  We discuss a generalized dominance ordering for irreducible representations
of the symmetric group $S_{n}$ with the aim of distinguishing the corresponding
states in the 1/2-BPS sector of $U(N)$ Super Yang-Mills theory when a certain
finite number of Casimir operators are known. Having knowledge of a restricted
set of Casimir operators was proposed as a mechanism for information loss in
this sector and its dual gravity theory in AdS$_{5}\times S^{5}$. It is
well-known that the states in this sector are labeled by Young diagrams with
$n$ boxes. We propose a generalization of the well-known dominance ordering of
Young diagrams. Using this generalization, we posit a conjecture to determine
an upper bound for the number of Casimir operators needed to distinguish
between the 1/2-BPS states and thus also between their duals in the gravity
theory. We offer numerical and analytic evidence for the conjecture. Lastly, we
discuss implications of this conjecture when the energy $n$ of the states is
asymptotically large.
