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TWO GENERAL COUNTING METHODS FOR ARRANGEMENTS
THREE GENERATING FUNCTIONS
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2-colorings a-z cut a-z flow adjacent arrangements balls biconnected components binomial binomial coefficients bipartite graph coefficient color column combinatorial corners counting problems cube darkened squares digits directed graph disjoint distinct objects distribute entry equation equivalent Euler circuit Example Exercise exponential generating function Find a recurrence formula four graph coloring graph G graph in Figure graph theory Grundy number Hamiltonian circuit identical objects induction integer integer solutions isomorphic kernel labeled least letters lexicographic order matching mathematical mathematical induction matrix maximal flow minimal n-digit number of different number of vertices obtain partitions pattern inventory permutation pick pile planar graph player polynomial position possible Prim's algorithm probability progressively finite game proof Prove recurrence relation rotation secret code Section shortest path Show shown in Figure solve spanning tree subset Suppose symmetries Theorem tour triangle undirected unoriented vertex vertices of degree