164 MAJOR P. A. MACMAHON: MEMOIR ON THE 



indicates that 6 is in the first row corresponding to a, 

 5 second /3, 

 4 ,, first ,, ,, ,, , 

 3 third y, 

 2 second ft, 

 I first . 



Hence the arrangement 



641 



52 

 3 



The transformation is quite general ; thus from 



12 9 7 4 

 11 8 6 2 



10 5 3 1 



we pass to 



a fl y a /8 /8yy/3y 



12 8 4 11 7 10 6 3 9 2 5 1, 

 a permutation in which the descending orders indicated by 



12 11 10 9 

 8765 



4321 

 are in evidence. 



17. The first question is the enumeration of the partitions where the graph and the 

 set of unequal numbers are given. 



Let the graph contain a, b, c, ... nodes in the successive rows, and for a given set 

 of a + b + c+... unequal numbers, let 



(abc...;) 



denote the number of partitions. 

 Observe that above we found 



(321 ;) = 16. 

 First we note that 



*Next take a graph of two rows. In any such graph 



Compare " Probleme des deux files de soldats," ' Theories des Nombres,' tome 1, p. 86, par EDOUARD LUCAS. 



