Freaks City

Consider two sequences of non-negative integers \(p_1 \ge p_2 \ge \dots \ge p_n\) and \(q_1 \ge q_2 \ge \dots \ge q_m\) satisfying \(\sum_{i=1}^n p_i = \sum_{i=1}^m q_i\). The Gale-Ryser theorem states that a simple bipartite graph exists with left vertices having degrees \(p_1, p_2, \dots, p_n\) and right vertices having degrees \(q_1, q_2, \d ...