Arrangements of Fifteen Symbols* 5T1 



Determine the number of combinations that can be made out of n 

 symbols, p symbols in each ; with this limitation, that no com- 

 bination of q symbols which may appear in any one of them 

 shall be repeated in any other. 



This question, which essentially involves a developed theory 

 of partitions, is more difficult than would at first appear ; and it 

 has not yet received anything like an approach to a complete 

 general investigation, although it has given rise to some able 

 papers on cognate subjects by Professor Sylvester, Mr. Cay ley, 

 &c. in the Philosophical Magazine and other scientific journals. 

 The Rev. T. P. Kirkman, who, like Professor Sylvester, has gone 

 somewhat elaborately into the subject of partitions, and has 

 brought considerable ingenuity to bear upon his researches, has 

 made the largest contributions towards the solution of the pro- 

 blem referred to. His early investigations in this particular 

 field of inqiiiry led him to construct the following curious 

 triadic problem, which was proposed amongst the queries given 

 in the * Lady's and Gentleman's Diary ' for 1850 : — 



Fifteen young ladies in a school walk out three abreast for seven 

 days in succession : it is required to arrange them daily so that 

 no two shall walk twice abreast. 



Two solutions, one of them by the talented proposer, were 

 printed in the ' Diary ' for 1851 ; but in these the results only 

 were exhibited. Since that time the question has found its way 

 into general society, and become somewhat noted as a fashionable 

 puzzle, while, more scientifically considered, it has not failed to 

 attract the attention of several eminent mathematicians. 



Professor Sylvester, at the end of his paper " On a Four-valued 

 Function," printed in the Philosophical Magazine for June last, 

 page 520, has made some passing allusions to those mathe- 

 maticians who, in common with himself, have contributed to 

 the subject under consideration. On a recent perusal of this 

 interesting paper I could not help noticing the summary cha- 

 racter of these allusions, which first suggested to my mind the 

 propriety of making the present communication with the view of 

 pointing out the fact that the Rev. T. P. Kirkman originated this 

 particular problem, and that it first appeared in the ' Diary.' I 

 have at the same time been induced to give a systematic and 

 comprehensive investigation of everything relating to it, in the 

 * Diary l for 1862, just published. In the investigation there 

 given, it is shown that every solution to the problem must be 

 contained in one or other of the three following systems : — 



