Séminaire Lotharingien de Combinatoire, B42h (1999), 10 pp.
David P. Little
An Extension of Franklin's Bijection
We are dealing here with the power series expansion of the product
F(m,q)=(1-qm+1)(1-qm+2)(1-qm+3)... This expansion may be readily obtained from an
identity of Sylvester and the latter, in turn, may be given a relatively
simple combinatorial proof. Nevertheless, the problem remains to give a
combinatorial explanation for the massive cancellations which produce the
final result. The case m=0 is clearly explained by Franklin's proof of the
Euler Pentagonal Number Theorem. Efforts to extend the same mechanism of proof
to the general case m>0 have led to the discovery of an extension
of the Franklin involution which explains all the components of
the final expansion.
Received: December 16, 1998; Accepted: February 11 1999.
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