The ranks of partitions modulo 2 (Q1356482)

\textit{F. J. Dyson} [Eureka 8, 10--15 (1994)] defined the rank of a partition \(\pi\) by \(\text{rank} (\pi):=\pi_0-\#\pi\). Setting \(N(r,m,n): =\# \{\pi: (\omega(\pi)=n\), \(\text{rank}(\pi) \equiv r\bmod m\}\) he found empirically that \[ N(0,5,5n+4)= N(1,5,5n+4)= N(2,5,5n+4)\tag{1} \] and \[ N(0,7,7n +5)=N(1,7,7n+5) =N(2,7,7n+5)= N(3,7,7n+5).\tag{2} \] It is easy to see that \(N(r,m,n)= N(-r,m,n)\) and Dyson's observations provide concrete realisation of Ramanujan congruences \(p(5n+4)\equiv 0\bmod 5\) and \(p(7n+5)\equiv 0\bmod 7\). However, Dyson also observed that the rank does not serve to realise the third Ramanujan congruence, \(p(11n+6)\equiv 0\bmod 11\) in the same way. All Dyson's observations were proved by \textit{A. O. L. Atkin} and \textit{P. Swinnerton-Dyer} [Proc. Lond. Math. Soc. (3) 4, 84--106 (1954; Zbl 0055.03805)]. They used heavy analytical methods in their proof and, since (1) and (2) are purely combinatorial statements, it would be satisfactory if there were combinatorial (bijective) proofs of these statements. Here the author gives a bijective proof of the following theorem: \[ N(0,2,2n)<N(1,2,2n)\text{ if }n>0, \] \[ N(0,2,2n+1) >N(1,2,2n+1) \text{ if }n\geq 0. \] Set \(R(\varepsilon)= \{\text{partitions }\pi: \omega(\pi)>0\), \(\omega(\pi)+ \pi_0+\#\pi \equiv\varepsilon \bmod 2\}\). Then he proves the theorem (with weak inequalities) by constructing an injective map \(\Phi:R(0)\to R(1)\). The theorem itself is then established by noting that none of the partitions \(1+1+\cdots+1\), which lie in \(R(1)\), lies in the image of \(\Phi\). He uses a variation of Garsia and Milne's involution principle [\textit{A. M. Garsia} and \textit{S. C. Milne}, J. Comb. Theory, Ser. A 31, 289--339 (1981; Zbl 0477.05009)].