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Is there a known bijective proof of Ramanujan's congruence for the partition function modulo 5? E.g., is there a construction that for every $n$ congruent to 4 mod 5 gives a permutation of the partitions of $n$ that increases the Dyson rank of each partition by 1 mod 5?

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up vote 3 down vote accepted

It's hard to be absolutely certain of course, but I would say no.

The most recent reference I can find mentioning the absence of such a proof is this article by Bessenrodt and Pak (2003). I also got some negative answers from modular form specialists to whom I asked the very same question fairly recently...

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