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 $\lambda$ where $f^\lambda \leq n^d$ for fixed $d$.