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1-cycles arrangements binomial coefficients binomial theorem Burnside's theorem called Catalan numbers choose colors Combinatorial Theory conjugacy class contains count the number cube cycle notation defined degree denote the number diagonal digits disjoint cycles divides elements in G equal equation Euler path exactly factors Fibonacci Fibonacci numbers formula four function Hamiltonian circuit Hamiltonian path identity induction integers inverse isomorphic Let us consider means multinomial coefficients multiply multiset mutual strangers n-gon n-set notation number of different number of distinct number of edges number of elements number of partitions number of paths number of permutations number of vertices odd number odd permutation one-to-one correspondence orbit pair paths from 0,0 pigeonhole principle Polya's theorem polynomial possible previous problem previous theorem Proof Let Prove redundant rotation rule of product Show Stirling numbers subgroup Sylvester's formula symbol Theorem 90 total number transpositions triangle vertex weight