M \.loi; !'. A. MvrMAHON ON THK COMPOSITIONS OF NUMBERS. 71 



The general law is clear; the letters ", l>, <, d are always in order in the 

 denominators and the sign of a fraction depends upon the number of factors in its 

 denominator. 



We can thus calculate the number of permutations appertaining to each of the 

 2"" 1 compositions of n. 



It has been established independently, by the aid of the zig-zag graph, that these 

 numbers M- / \ 



are equal in four's or in two's. 



Art. 8. The sum of the numbers N (...) is of course n! 

 The details of the above results for 



n = 2, 3, 4, 5, 6 

 are given for easy reference. 



N(2) = 1 1 



N(l') = 1 1 



2 = 2! 



N(3) =N(1 3 ) = 1 2 



N(21) = N(12) = 2 4 



6 = 3! 



N (4) = N (I 4 ) = 1 2 



N(31) = N(13) = N(21') = N(1'2) = 3 12 



N(22) = N(121) =5 10 



24 = 4! 



N(5) =N(1 & ) = 1 2 



N(41) = N(14) = N(21 3 ) =N(1 3 2) = 4 16 



N(121) = N(1 !I 21)= 9 36 



N(31 8 ) =N(l a 3) = 6 12 



N(2*l) =N(12*) = 16 32 



N(131) =N(212) = 11 22 



120 = 5! 



