Iterated Lusztig-Vogan Bijection and Distinguished Weights

The distinguished weights form a subset of the weight lattice and are closely tied to the notion of $p$-cells. These weights are defined via iterations of the Lusztig-Vogan bijection. We prove that all distinguished weights exhibit an anti-symmetry under the composition of reversal and negation. We show that the distribution of these weights follows a polynomial asymptotic, with a leading coefficient relating to the telephone numbers. As an explicit computation, we determine all the distinguished weights for $n\le 4$.