In the theory of permutation patterns, the influence of the presence and absence of ordered subsequences on the structural properties of permutations is studied. In the thesis, the foundations of the theory are presented from fundamental definitions to contemporary results. For patterns of length 3, it is proved that they all form a single Wilf class, which is enumerated by the Catalan numbers. Special emphasis is placed on patterns of length 4, wherein the classification of Wilf classes, the open problem of the pattern 1324, and Bóna’s proof of an upper bound are treated. Finally, the Stanley–Wilf theorem is presented, which shows that the number of avoiding permutations grows at most exponentially. The aim of the thesis is to present the key concepts and results and to demonstrate the significance of this area in contemporary combinatorics.
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