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Permutations are fundamental objects in mathematics, biology and many other fields. Different presentations reveal different aspects of permutations, for instance, the one-line notation describes how their patterns go up and down while the reduced words highlight their construction through transpositions. I will talk about certain tableaux, posets, and polytopes from permutations and the interplay between them. This talk is based on joint works with Banaian, Chepuri, Gunawan, Russel and Tenner.