Séminaire Lotharingien de Combinatoire, 84B.44 (2020), 12 pp.
Sara C. Billey, Matjaž Konvalinka and Joshua P. Swanson
On the Distribution of the Major Index on Standard Young Tableaux
Abstract.
The study of permutation and partition statistics is a classical topic
in enumerative combinatorics. The major index
statistic on permutations was introduced a century ago by Percy MacMahon in his
seminal works. In this extended abstract, we study the well-known generalization of the major index
to standard Young tableaux. We present several new results. In one
direction, we introduce and study two partial orders on the standard Young tableaux
of a given partition shape, in analogy with the strong and weak
Bruhat orders on permutations. The existence of such
ranked poset structures allows us to classify the realizable major
index statistics on standard tableaux of arbitrary straight shape
and certain skew shapes, and has representation-theoretic consequences,
both for the symmetric group and for Shephard-Todd groups. In a different
direction, we consider the distribution of the major index on
standard tableaux of arbitrary straight shape and certain skew
shapes. We classify all possible limit laws
for any sequence of such shapes in terms of a simple auxiliary
statistic, aft, generalizing earlier results of
Canfield-Janson-Zeilberger, Chen-Wang-Wang, and others.
We also study unimodality, log-concavity, and local limit properties.
Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.
The following versions are available: