We prove that the expected number of braid moves in the commutation class of the reduced word $(s_1 s_2 \cdots s_{n-1})(s_1 s_2 \cdots s_{n-2}) \cdots (s_1 s_2)(s_1)$ for the long element in the symmetric group $\mathfrak{S}_n$ is one. This is a variant of a similar result by V. Reiner, who proved that the expected number of braid moves in a random reduced word for the long element is one. The proof is bijective and uses X. Viennot's theory of heaps and variants of the promotion operator. In addition, we provide a refinement of this result on orbits under the action of even and odd promotion operators. This gives an example of a homomesy for a nonabelian (dihedral) group that is not induced by an abelian subgroup. Our techniques extend to more general posets and to other statistics.

Studies with older children and adults have found that physically engaging video games (i.e., Exergames) that promoteboth cognitive control and physical activity improve executive function (EF) skills; yet, children below school age remainunderstudied with regard to the impact of Exergames on EF. Additionally, research on the extent of the impact of Ex-ergames resulting in prolonged changes, and whether training generalizes to EF-related behaviors in a real-world contextremains scarce. This study examined the short- and long-term changes in EF of 4- to 5-year-olds after participation in two20-minute Exergame sessions. Results indicate that Exergame training improved performance on EF tasks and resultedin higher teacher ratings of EF in the classroom compared to a sex-/classroom-/age-matched control group. The improve-ments in EF persisted over a one-month period. This study provides novel insights into the short-term and long-termeffects of Exergame training on executive function in preschool-aged children.