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Polynomial generating functions in combinatorics give rise to random variables for related discrete probability distribution functions. When the generating functions count families of objects indexed by a parameter going to infinity, we can ask what the limiting distribution looks like. For many familiar combinatorial objects, the limiting distribution is the normal distribution. Examples include permutations, set partitions, and multisets with certain statistics. After reviewing the concept of asymptotic normality, we will give simple necessary and sufficient conditions under which the major index on standard tableaux of fixed partition shape is asymptotically normally distributed. The major index statistic on tableaux has a close connection to the symmetric group action on the coinvariant algebra, so our results have implications for the irreducible multiplicities. We extend the main result to certain skew shapes to obtain a uniform generalization of analogous results of (Canfield--Janson--Zeilberger 2011) for words and (Chen--Chen--Wang 2008) for $q$-Catalan coefficients. A byproduct of the proof is an explicit formula for all higher cumulants or, equivalently, moments for the major index on tableaux.

This talk is based on joint work with Matjaz Konvalinka and Joshua Swanson.