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Applying the same procedure to any positive integer <math>n\!</math> produces an expression called the ''doubly recursive factorization'' of <math>n.\!</math> &nbsp; If <math>\mathbb{M}</math> is the set of positive integers, <math>\mathcal{L}</math> is the relevant set of factorization expressions, and the mapping defined by the factorization process is denoted <math>\operatorname{drf} : \mathbb{M} \to \mathcal{L},</math> then the doubly recursive factorization of <math>n\!</math> is denoted <math>\operatorname{drf}(n).\!</math>

The pattern of indices and exponents exhibited in this example is called a ''doubly recursive factorization'', or ''DRF''.  Applying the same procedure to any positive integer <math>n\!</math> produces an expression called the DRF of <math>n.\!</math> &nbsp; If <math>\mathbb{M}</math> is the set of positive integers, <math>\mathcal{L}</math> is the set of DRF expressions, and the mapping defined by the factorization process is denoted <math>\operatorname{drf} : \mathbb{M} \to \mathcal{L},</math> then the doubly recursive factorization of <math>n\!</math> is denoted <math>\operatorname{drf}(n).\!</math>

The forms of DRF expressions can be mapped into either one of two classes of graph-theoretical structures, called ''riffs'' and ''rotes'', respectively.

The forms of DRF expressions can be mapped into either one of two classes of graph-theoretical structures, called ''riffs'' and ''rotes'', respectively.