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Séminaire Lotharingien de Combinatoire, B42h (1999), 10 pp.

# David P. Little

#
An Extension of Franklin's Bijection

**Abstract.**
We are dealing here with the power series expansion of the product
*F*(*m*,*q*)=(1-*q*^{m+1})(1-*q*^{m+2})(1-*q*^{m+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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