Base.Matrix — MethodMatrix(f::DiscreteFunction) returns an n-by-n zero-one matrix in which there is a 1 in position i,j exactly when f(i)==j.
DiscreteFunctions.DiscreteFunction — TypeDiscreteFunction is a function from {1,2,...,n} to itself. It can be created by DiscreteFunction(list) where list is a one-dimensional array of positive integers. Alternatively, it can be created using positive integer arguments.
The following are equivalent:
DiscreteFunction([1,4,2,3])
DiscreteFunction(1,4,2,3)DiscreteFunctions.DiscreteFunction — MethodDiscreteFunction(A) creates a new discrete function based on the square matrix A. It is required that each row of A have exactly one nonzero element. Note that if A=Matrix(f) then DiscreteFunction(A)==f.
Base.:+ — MethodIf f and g are of type DiscreteFunction, then f+g is their disjoint sum.
Base.sqrt — Methodsqrt(g::DiscreteFunction) returns a function f such that f*f==g or throws an error if no such function exists. This method uses integer linear programming.
DiscreteFunctions.IdentityFunction — MethodIdentityFunction(n) creates the identity DiscreteFunction on the set {1,2,...,n}.
DiscreteFunctions.RandomFunction — MethodRandomFunction(n) creates a random DiscreteFunction on {1,2,...,n}.
DiscreteFunctions.all_functions — Methodall_functions(n) returns a generator that produces all DiscreteFunctions on {1,2,...,n}
DiscreteFunctions.all_sqrts — Methodall_sqrts(f::DiscreteFunction) returns an array consisting of all g such that g*g==f.
DiscreteFunctions.all_squares — Methodall_squares(n) returns a list of all function on [n] that are perfect compositional squares.
DiscreteFunctions.has_inv — Methodhas_inv(f::DiscreteFunction) tests if f is invertible.
DiscreteFunctions.has_sqrt — Methodhas_sqrt(f::DiscreteFunction) checks if there is a function g such that g*g==f. Returns true if so and false otherwise.
DiscreteFunctions.image — Methodimage("f::DiscreteFunction") returns a Set containing all the output values of f.
DiscreteFunctions.is_permutation — Methodis_permutation(f::DiscreteFunction) returns true if f is a bijection on its domain.
DiscreteFunctions.quick_sqrt_test — Methodquick_sqrt_test(f::DiscreteFunction) is a quick check of a necessary condition for f to have a square root. If this returns false then f does not have a square root. If it returns true, it might.
DiscreteFunctions.random_square — Methodrandom_square(n) returns a DiscreteFunction on n elements that has a compositional square root.
DiscreteFunctions.reduced_eigvals — Methodreduced_eigvals(f::DiscreteFunction) returns the nonzero eigenvalues of f's matrix.
DiscreteFunctions.sources — Methodsources(f::DiscreteFunction) returns a list of all inputs to f that are not the output of f. That is, this is a list of the vertices with out-degree 0 in the graph of f.
LinearAlgebra.eigvals — Methodeigvals(f::DiscreteFunction) returns the eigenvalues of Matrix(f).
Permutations.cycles — Methodcycles(f::DiscreteFunction) returns a list of the cycles in f.
Permutations.fixed_points — Methodfixed_points(f::DiscreteFunction) returns a list of fixed points of the function, i.e., those values x such that f(x)==x.