the number of standard tableaux of a given shape and major index $r$ mod $n$ give the irreducible multiplicities of certain induced or restricted representations. we give simple necessary and sufficient conditions classifying when this number is zero. this result generalizes the $r=1$ case due essentially to klyachko (1974) and proves a recent conjecture due to sundaram (2016) for the $r=0$ case. indeed, we prove a stronger asymptotic uniform distribution result for ``almost all'' shapes. we'll discuss aspects of the proof, including a representation-theoretic formula due to desarmenien, normalized symmetric group character estimates due to fomin-lulov, and new techniques involving ``opposite hook lengths'' for classifying $\lambda \vdash n$ where $f^\lambda \leq n^d$ for fixed $d$.