Exact(21)
Theorem 2.7 The inefficient but minimal presentation P M defined in (7) has a set of generating functions.
Theorem 2.12 The efficient presentation P M in (7) has a set of generating functions in terms of Stirling numbers as given in (13).
Corollary 3 The presentation P M in (12) has a set of generating functions in terms of Stirling numbers as given in (18).
Remark 2 From Corollary 2 and by considering Theorem 1, we can easily deduce that the set of generating functions for a minimal but inefficient presentation can be presented as a single element.
By taking i = 2 and k − l is an odd positive integer, we get a set of generating functions in terms of Stirling numbers for the inefficient but minimal presentation of the form as defined in (7).
Corollary 2.6 For any prime p, the presentation P M = [ y, x ; y ( p + 1 ) [ ( p + 1 ) p − 1 p ] + 1 = y, x p + 1 = x, y x = x y p + 1 ]. has a set of generating functions.
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