An Involution Proof of the Alladi-Gordon Key Identity for Schur's Partition Theorem

摘要

The Alladi-Gordon identity $sum_{k=0}^{j}(q^{i-k+1};q)_k, {j rack k} q^{(i-k)(j-k)}=1$ plays an important role for the Alladi-Gordon generalization of Schur's partition theorem. By using Joichi-Stanton's insertion algorithm, we present an overpartition interpretation for the Alladi-Gordon key identity. Based on this interpretation, we further obtain a combinatorial proof of the Alladi-Gordon key identity by establishing an involution on the underlying set of overpartitions. Author Biography James J.Y. Zhao, Dongling School of Economics and ManagementUniversity of Science and Technology Beijing Beijing 100083P.R. China Lecturer