Major Index Asymptotics

Abstract: Kendall's tau independence test relies on knowing the distribution of the inversion statistic on permutations. The inversion statistic is well known to probabilists as being asymptotically normal, and its probability generating function is very well known to combinatorialists. MacMahon classically introduced another statistic on permutations, the major index, and showed it to be equidistributed with the inversion statistic by computing its probability generating function. The first part of the talk will summarize a new approach to a result of Baxter--Zeilberger concerning the joint independent asymptotic normality of the pair (inv, maj) using a generating function identity of Roselle, answering a question of Romik and Zeilberger. The argument involves a blend of tools from combinatorics and probability: characteristic function estimates, generating function manipulatorics, and Mobius inversion on the partition lattice. The major index statistic has been generalized from permutations to standard Young tableaux and has been used to answer several representation theoretic questions. The second part of the talk will discuss joint work with Sara Billey and Matjaz Konvalinka classifying all possible limiting normalized distributions of the major index statistic on tableaux for a wide class of partition shapes using cumulant estimates. This result provides a common generalization and strengthening of a series of results due to Mann--Whitney, Feller, Diaconis, Chen--Wang--Wang, and others.

 

The talk will include a gentle introduction to the necessary combinatorics.