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 fλ <= nd for fixed d.

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

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