Séminaire Lotharingien de Combinatoire, 78B.50 (2017), 9 pp.
Joshua P. Swanson
Standard Tableaux and Modular Major Index
We provide simple necessary and sufficient conditions for the
existence of a standard Young tableau of a given shape and major index
r mod n, for all r. Our 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. A byproduct of the proof is an
asymptotic equidistribution result for ``almost all'' shapes. The
proof uses a representation-theoretic formula involving Ramanujan sums
and normalized symmetric group character estimates. Further estimates
involving ``opposite'' hook lengths are given which are well-adapted
to classifying which partitions λ of n have
fλ <= nd for fixed d.
Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.
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