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Partition congruences and the Andrews-Garvan-Dyson crank

  1. Karl Mahlburg?
  1. Department of Mathematics, University of Wisconsin, 418 Van Vleck Hall, E B, 480 Lincoln Drive, Madison, WI 53706
  1. Communicated by George E. Andrews, Pennsylvania State University, University Park, PA, August 4, 2005 (received for review June 1, 2005)

Abstract

In 1944, Freeman Dyson conjectured the existence of a “crank” function for partitions that would provide a combinatorial proof of Ramanujan's congruence modulo 11. Forty years later, Andrews and Garvan successfully found such a function and proved the celebrated result that the crank simultaneously “explains” the three Ramanujan congruences modulo 5, 7, and 11. This note announces the proof of a conjecture of Ono, which essentially asserts that the elusive crank satisfies exactly the same types of general congruences as the partition function.

Footnotes

    • ?? E-mail: mahlburg{at}math.wisc.edu.

    • Author contributions: K.M. performed research and wrote the paper.

    • See Commentary on page 15277.

    • Received June 1, 2005.

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