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XIX. — A new Algebra, by means of which Permutations can be transformed in a 

 variety of ivays, and their properties investigated. By T. B. Sprague, M.A., 

 F.R.S.E. 



(Read 6th March 1893.) 



In the course of investigations that I have lately been making into the transformation 

 of permutations, I have found it convenient to employ several new symbols of operation, 

 which combine with each other according to laws that differ materially from the ordinary 

 algebraical laws ; and it is my object in this paper to explain those laws, and give a few 

 examples of the manner in which various propositions relating to permutations may be 

 demonstrated by means of my symbols. 



Taking any permutation of the first n natural numbers, which we may denote by P, 

 let t denote the operation of taking the first number in the permutation, and putting it 

 last, thus forming a new permutation ; for instance, if n = 5, and the permutation is 

 13452, so that P = (13452) ; then t? - (34521). Also ttP = fP = (45213) ; Z 3 P = (52134) ; 

 £ 4 P = (21345); £ 5 P = (13452) = P. This result shows that in this case t 5 =l; and, in 

 general, it is obvious that, if the operation t is performed n times on a permutation, we 

 get the original permutation again, so that t n = 1. 



Let s denote the operation of forming a new permutation by subtracting 1 from each 

 of the constituents of P, but substituting n for where it occurs ; so that, taking the 

 same example as before, sP = (52341); s 2 P = (41235); s 3 P = (35124); s 4 P = (24513); 

 s 5 P = (13452). Here we see that s 5 P is the same as P, or in this case s 5 = 1 ; and, in 

 general, it is obvious that, if the operation s be performed n times on a permutation, the 

 original one is reproduced, so that s n = 1. 



A little consideration shows us that powers of s and t may be combined according to 

 the laws of algebraical multiplication ; so that, for instance, sV = s h+k ; t h t k = t + ; s h t k = 

 t k s'' ; &c. It is obvious how we must interpret s' 1 and t' 1 : the effect of the former is to 

 add 1 to each constituent of P, replacing (n + 1) by 1 ; and the effect of the latter is to 

 transpose the last constituent in P to the first place. Thus: — s -1 (13452) = (24513), 

 r a (13452) = (21345). 



If now we form all possible permutations of the form sVP, by giving h and Jc all the 

 values 0, 1, 2, . . . . n—1, we shall get n 2 permutations, which I call a 

 "set". Thus, taking our former example, we get from 13452 a set contain- 523415934 

 ing 25 permutations. These are all contained in the annext scheme, where 412354123 

 any five consecutive figures form one of the permutations, and there are 04^09451 

 thus 5 permutations in each line. Any desired permutation can at once be 



read off from it ; for instance, 



s 2 £ 3 P = (35412); s^ 2 P = (12435). 

 VOL. XXXVII. PART II. (NO. 19). 3 N 



