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- W2963457572 abstract "Andrews, Lewis and Lovejoy introduced the partition function PD(n) 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 PD(3n+2) is divisible by 3 and showed that the generating function of PD(3n) can be expressed as an infinite product of powers of (1−q2n+1) times a function F(q2). We obtain a Ramanujan type identity which implies the congruence for PD(3n+2). We also find an explicit formula for F(q2), which leads to a formula for the generating function of PD(3n). A formula for the generating function of PD(3n+1) is also obtained. Our proofs rely on Chanʼs identity on Ramanujanʼs 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 PD(3n+2)." @default.
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- W2963457572 date "2013-09-01" @default.
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- W2963457572 title "On the number of partitions with designated summands" @default.
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- W2963457572 doi "https://doi.org/10.1016/j.jnt.2013.02.010" @default.
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