An amusing passage from Doron Zeilberger’s PCM article on enumerative combinatorics (which I’ve finally gotten around to reading):

According to the modern approach, pioneered by Polya, Tutte, and Schutzenberger, generating functions are neither “generating,” nor are they functions.

Discuss!

(The second post on Tutte(1, 0) is forthcoming; I need to get in touch with the student I’m working with before I post it. I’m also working on a post on category theory in combinatorics, which will hopefully be up on Monday.)

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June 29, 2009 at 05:37 |

Doron Zeilberger is fond of making rhetorical points like that. Probably what he means is that one shouldn’t think of an analytic function as giving rise to a Taylor series expansion; the real importance is in the combinatorial objects those coefficients describe, which is more basic than either the analytic properties of the corresponding generating function (its “functional” property) or the way in which one can deduce formulae for its coefficients from those analytic properties (its “generating” property).

June 29, 2009 at 17:22 |

That seems about like the point he was making, and it is a good point, and one which I’ll probably have more to say about in my next post. (Of course, the analytic properties of the function — when they exist — shouldn’t be disregarded either!)

That doesn’t make it any less amusing to me for its resemblance to the Mike Myers sketches, though.