An Extension of Franklin's Bijection

David P. Little

Séminaire lotharingien de combinatoire, Tome 42 (1998-1999)

Résumé

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.