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

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Keywords: the Alladi-Gordon key identity, Joichi-Stanton's insertion algorithm, Schur's celebrated partition theorem, overpartitions

Abstract

The Alladi-Gordon identity sumk=0j(qi−k+1;q)k,jbrackkq(i−k)(j−k)=1\sum_{k=0}^{j}(q^{i-k+1};q)_k\, {j \brack k} q^{(i-k)(j-k)}=1sumk=0j(qi−k+1;q)k,jbrackkq(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 Management University of Science and Technology Beijing Beijing 100083 P.R. China

Lecturer

Dongling School of Economics and Management

University of Science and Technology Beijing

Beijing 100083

P.R. China

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How to Cite

Zhao, J. J. (2013). An Involution Proof of the Alladi-Gordon Key Identity for Schur’s Partition Theorem. The Electronic Journal of Combinatorics, 20(1), #P63. https://doi.org/10.37236/2826