7-j MAJOR P. A. MACMAHON ON THE COMPOSITIONS OF NUMBERS. 



N(6) = N(i") = i 2 



N(51) =N(15) = N(21 4 ) = N(1 4 2) =5 20 



N(42) =N(24) = N(1 8 21) =N(121 3 ) = 14 56 



N(3 2 ) = N(1 2 21 2 ) = 19 38 



N(41 a ) =N(1 2 4) =N(31 3 ) = N(1 3 3) =10 40 



N(141) = N(21 2 2) = 19 38 



N(321) = N(123) = N(2 2 1 2 ) = N(1 2 2 2 ) =35 140 



N (312) = N (213) = N (1 2 31) = N (131*) = 26 104 



N(132) = N(231) = N(2121) = N(1212) = 40 160 



N(2 S ) =N(12 2 1) =61 122 



720 = 6! 



Art. 9. Some simple summations are obtainable from elementary considerations. 

 In regard to the permutations of the first n integers, let 



' ''"'.* 2N (*...), 



where <n, denote the sum of all numbers N (...), such that s is the first number in 

 the specifying composition. Take any s+1 of the numbers 



1, 2, 3,...w, 



and arrange them from left to right in such wise that the first * numbers are in 

 descending order and the s+ 1 th number greater than the s th ; this can be done in 



n \ 



+ 1 J ways; 



the remaining n s 1 numbers can be arranged in (n s 1)! ways, so that, placing 

 them to the right of the former, we arrive at the result 



SN (...)=- n! 

 Art. 10. Again, denoting by 



the sum of all numbers N (...) of which the specifying compositions commence with 

 exactly * 1 units, the consideration of the properties of conjugate zig-zag graphs 

 establishes that SN (!"'...) = SN (...), 



with a single exception where s = n ; e.g., 



[...)= SN(2...) = 2^, 

 and so on. 



No restriction is placed upon the number next to the unit in this case, 

 t Here the number following the unit must be > 1. 



