Congruence properties for the partition function

@article{Ahlgren2001CongruencePF,
  title={Congruence properties for the partition function},
  author={Scott Ahlgren and Ken Ono},
  journal={Proceedings of the National Academy of Sciences of the United States of America},
  year={2001},
  volume={98},
  pages={12882 - 12884}
}
  • S. Ahlgren, K. Ono
  • Published 23 October 2001
  • Mathematics
  • Proceedings of the National Academy of Sciences of the United States of America
Eighty years ago, Ramanujan conjectured and proved some striking congruences for the partition function modulo powers of 5, 7, and 11. Until recently, only a handful of further such congruences were known. Here we report that such congruences are much more widespread than was previously known, and we describe the theoretical framework that appears to explain every known Ramanujan-type congruence. 
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References

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Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin,…
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SummaryNew statistics on partitions (calledcranks) are defined which combinatorially prove Ramanujan's congruences for the partition function modulo 5, 7, 11, and 25. Explicit bijections are given…
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j(^r) = E c(n)xn = (1 +240 0 u3(n)xn) /xf24(x) 744, where 03(n) = d3. dln Then p(n) is just the number of unrestricted partitions of n, and c(n) is the Fourier coefficient of Klein's modular…
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Congruence properties of partitions
Congruences like Ramanujan's for powers of the partition function
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