Thresholds for patterns in random compositions and random permutations

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We explore how the asymptotic structure of a random permutation of [n] with minversions evolves, as m increases, establishing thresholds for the appearance and disappearance of any classical, consecutive or vincular pattern. Our investigation beginswith exploring how the asymptotic structure of a random n-term weak integer composition of m evolves, as m increases from zero. The primary focus of our investigationinto compositions is establishing thresholds for the appearance and disappearance ofsubstructures, particularly the appearance and disappearance of consecutive composition patterns. We are then able to transfer the established composition thresholdto establish the thresholds for classical, consecutive or vincular permutation patternsoccurring within our random permutation model.This thesis is based on the papers [12] and [13].

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