Major MacMahon, Note on the parity of the number, etc. 281 



Note on the parity of the number which enumerates the 

 partitions of a number. By Major P. A. MacMahon. 



\_Read 25 October 1920.] 



In a letter received by rae from the late S. Ramanujan about 

 a year ago he stated that it was his intention to calculate the 

 number of the partitions of 1000 by the direct approximate 

 formula which had been successfully used by him and G. H. Hardy 

 in the calculation of the case n = 200. He enquired, at the same 

 time, if I knew of any simple way of ascertaining whether the 

 sought number is even or uneven as this information would be of 

 importance to him. This note has arisen in consequence of this 

 enquiry. 



I shew, in particular, that in the case of the partitions of 1000 

 the parity can be found, in a few minutes, from certain congruence 

 relations. 



It is easy to derive, from the theory of the self-conjugate 

 partitions of n, that 



^— — = n"" (1 + q-"^-^) mod 2. 



Y[m (1 _ gm) 1 

 1 



Thence 



1 H'" {(1 + cf^'') (1 + f''-^) (1 - q'"^)] 



1 



mod 2. 



n'«(i-g"^) n™(i-5'"") 



1 1 



The numerator of the fraction on the right is one of Jacobi's 

 elliptic products which has the series expression 



1 4- g + g3 + ^6 + ^10 + . . . _|. ^ i« (+1) + . . . . 



Wherefore if ^„ denote the number of partitions of n 



•Zpnq"" = il + q+q' + q' + q'°+ ■■■) ^pnq"' mod 2. 

 Put pn = cbn mod 2. 



Comparison of the coefficients of like powers of q yields the four 

 relations 



a^n = a»i + «M-7 + ««-9 + "n-30 + an-34 + (^n-w + <^»i-75 + ^^n-\ii 



+ 0^71-132 + ^W-195 + 'j^n-aOS + Ct«-282 + C('ii_294 + • • • ) 



a4u+i = Cin + «»i-5 + «»- 11 + «"-6 + «n-38 + «w-63 + Clji-si + «n-n6 



+ a7l-140 + O^n-185 + «.n-215 + C^n-270 + f*n-306 + • • • > 

 (X4n+2 = dn-l + Ctn-2 + ««-i6 + O^n-is + '^n-47 + ^71- 52 + <^n-9i + «n-ioi 



+ flt„_]57 + CE.ji-166 + C^n-SSe + a„_047 + a„_331 + • • • > 



«4n+3 = «n + ^n-3 + ^n-13 + an-22 + O'n-ii + «'»i-67 + O-n-S! + O-fi-ios 



+ «n-i48 + O^n-175 + an-2aB + «'n-258 + a^-m + • • • • 



