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黑龙江快乐十分前三直:Almost partition identities
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In the past, what we call “almost partition identities” were immediate corollaries of classical theta series identities, such as Jacobi’s triple product. Consequently, they were always of this form: “The number of partitions of n from a certain class with a given partition statistic even” equals “the number of partitions of n from the same class with the given partitions statistic odd.” This paper opens possibilities for research in this area, relying on subtle results in basic hypergeometric series and mock modular forms.
An almost partition identity is an identity for partition numbers that is true asymptotically of the time and fails infinitely often. We prove a kind of almost partition identity, namely that the number of parts in all self-conjugate partitions of n is almost always equal to the number of partitions of n in which no odd part is repeated and there is exactly one even part (possibly repeated). Not only does the identity fail infinitely often, but also, the error grows without bound. In addition, we prove several identities involving the number of parts in restricted partitions. We show that the difference in the number of parts in all self-conjugate partitions of n and the number of parts in all partitions of n into distinct odd parts equals the number of partitions of n in which no odd part is repeated, the smallest part is odd, and there is exactly one even part (possibly repeated). We provide both analytic and combinatorial proofs of this identity.
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Author contributions: G.E.A. and C.B. designed research, performed research, and wrote the paper.
Reviewers: P.P., Johannes Kepler University Linz; and C.S., North Carolina State University.
The authors declare no conflict of interest.
Published under the PNAS license.