Minimum Decomposition on Maxmin Trees

Maxmin trees are trees that consist of nodes that are either local minimums or maximums. Such trees were first studied by Postnikov. Later Dugan, Glennon, Gunnells, and Steingrimsson introduced the concept of weight to these trees and proved a bijection between maximum weight maxmin trees and permutations, defining weights for permutations. In addition, the q-Eulerian polynomial $E_n(x, q)$ is defined which relates descents and weights of permutations. This polynomial was later proven to exhibit a stabilization phenomenon by Agrawal et al. Extracting the formal power series $W_d(t)$ from the stabilization of these coefficients, $W_d(t)$ was conjectured to partially correspond to A256193. In our paper, we introduce a process called minimum decomposition to help us better understand maxmin trees. Using minimum decomposition, we present a new way to calculate the weight of different maxmin trees and prove the bijection between the coefficients of $W_d(t)$ and A256193.