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Introduction The permanent of a matrix is one of those objects that looks almost identical to something familiar — the determinant — yet behaves in a completely different way. Where the determinant carries a sign on each permutation, the permanent does not. This small difference makes the permanent combinatorially rich and computationally hard. In this post we develop the theory of the permanent from scratch: its definition, its combinatorial interpretation, key identities and inequalities, its relationship to counting problems, and why computing it is fundamentally harder than computing the determinant. ...