A note on statistical averages for oscillating tableaux

TL;DR

This paper introduces a new statistic called the weight for oscillating tableaux, providing a simple quadratic formula for its average, and discusses potential for a more conceptual proof using the homomesy paradigm.

Contribution

It defines the weight statistic for oscillating tableaux and derives a simple quadratic formula for its average, connecting combinatorics with differential posets and symmetry concepts.

Findings

Average weight is a quadratic polynomial in size of lambda and n.

The proof is largely computational using differential posets.

Potential for a more conceptual proof via homomesy paradigm.

Abstract

We define a statistic called the weight of oscillating tableaux. Oscillating tableaux, a generalization of standard Young tableaux, are certain walks in Young's lattice of partitions. The weight of an oscillating tableau is the sum of the sizes of all the partitions that it visits. We show that the average weight of all oscillating tableaux of shape lambda and length 2n plus the size of lambda has a surprisingly simple formula: it is a quadratic polynomial in the size of lambda and n. Our proof via the theory of differential posets is largely computational. We suggest how the homomesy paradigm of Propp and Roby may lead to a more conceptual proof of this result and reveal a hidden symmetry in the set of perfect matchings.