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Séminaire Lotharingien de Combinatoire, 78B.50 (2017), 9 pp.

# Joshua P. Swanson

# Standard Tableaux and Modular Major Index

**Abstract.**
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
*f*^{λ} <= *n*^{d} for fixed *d*.

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

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