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Applying the same procedure to any positive integer <math>n\!</math> produces an expression called the ''doubly recursive factorization'' (DRF) of <math>n.\!</math>  This may be abbreviated as <math>\operatorname{drf}(n).\!</math>

Applying the same procedure to any positive integer <math>n\!</math> produces an expression called the ''doubly recursive factorization'' (DRF) of <math>n.\!</math>  This corresponding function from positive integers to DRF expressions may be indicated as <math>\operatorname{drf}(n).\!</math>

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

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

The ''riff'' of <math>123456789\!</math> is the following digraph:

* <math>\operatorname{riff}(123456789)</math> is the following digraph:

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{| align=center cellpadding="6"

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The ''rote'' of <math>123456789\!</math> is the following graph:

* <math>\operatorname{rote}(123456789)</math> is the following graph:

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{| align=center cellpadding="6"