A subset of this can be explained relatively simply: A permutation on n elements is a way to reorder lists of length n. One way to represent it is how it reord

A subset of this can be explained relatively simply: A permutation on n elements is a way to reorder lists of length n. One way to represent it is how it reorders 1, 2, …, n, but this doesn’t show you much structure. It turns out that one can group a permutation into independent cycles. One reason this is useful is because you might want to ask about applying the same permutation repeatedly and cycles are a way to do that. A cycle is the part of a permutation that can be described as a (circular) list where each element is transformed into the one following it. For example we might write the cycle (1257) to indicate the permutation that takes 1 to 2, 2 to 5, 5 to 7, and 7 to 1. It turns out that any permutation of a finite set can be written as a product of disjoint cycles (apart from the choice of which order to write down the cycles and the first element of each cycle). Another question is what happens if you “change your view” when applying a permutation. That is, instead of applying p to reorder some list, first apply a permutation q, then p, then undo q. For example instead of modifying p to make it work on a list backwards you could reverse the list, then apply p, then unreverse it. It turns out that this operation cannot change the ‘shape’ of p. That is, it can change the numbers in the cycle but if p was a 3-cycle, two 5-cycles, and a 1-cycle (an n-cycle is a cycle with n elements) then the combination [q, p, undo_q] will also be a 3-cycle, two 5-cycles, and a 1-cycle. Furthermore, there always exists a a to transform between any two cycles with the same shape. Now notice that the elements of Young’s lattice – partitions of integers (ways to find a multi set of numbers that add up to a given total) – correspond to the different possible shapes of permutations.