On the number of partitions with designated summands

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• 作者：William Y.C. Chen^a ; ^b ; ^{chen@nankai.edu.cn} ; Kathy Q. Ji^a ; ^{ji@nankai.edu.cn} ; Hai-Tao Jin^a ; ^{jinht1006@mail.nankai.edu.cn} ; Erin Y.Y. Shen^a ; ^{shenyiying@mail.nankai.edu.cn}
• 关键词：05A17 ; 11P83 ; 05A30
• 刊名：Journal of Number Theory
• 出版年：2013
• 出版时间：September, 2013
• 年：2013
• 卷：133
• 期：9
• 页码：2929-2938
• 全文大小：161 K

文摘

Andrews, Lewis and Lovejoy introduced the partition function as the number of partitions of n with designated summands, where we assume that among parts with equal size, exactly one is designated. They proved that is divisible by 3 and showed that the generating function of can be expressed as an infinite product of powers of times a function . We obtain a Ramanujan type identity which implies the congruence for . We also find an explicit formula for , which leads to a formula for the generating function of . A formula for the generating function of is also obtained. Our proofs rely on Chan?s identity on Ramanujan?s cubic continued fraction and identities on cubic theta functions. By introducing a rank for partitions with designated summands, we give a combinatorial interpretation of the congruence for .