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.Comment: 7 page
