In the thesis, we first examine inversions of permutations, their properties, the generating function for the number of permutations of the set [n] with i inversions, and Bruhat partial orders. Then, we present how permutations can be represented with permutation graphs and characterize permutation graphs using a cohesive vertex-set order. We show that caterpillars are the only trees that are permutation graphs, and prove that for every caterpillar graph, there are exactly two permutations that generate a permutation graph isomorphic to the caterpillar. Next, we explore competitivity graphs, sets of competitors, sets of eventual competitors, and an algorithm to calculate the sets of eventual competitors that does not require the construction of a competitivity graph. Finally, we apply the algorithm to a concrete example featuring real-world data.
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