528 Mr. A. Cay ley's Note on the Theory of Permutations. 



whatever can be derived (and derived in one manner only) 

 from the primitive arrangement by means of a rule such as is 

 furnished by the symbol in question*; and moreover that the 

 number of inversions requisite in order to obtain the permu- 

 tation by means of the rule in question, is always the smallest 

 number of inversions by which the permutation can be derived. 

 Let «, /3... be the number of letters in the components {scyz)^ 

 (w) {vw) &c., K the number of these components. The num- 

 ber of inversions in question is evidently a— 1 +/3 — 1 +&c., 

 or what comes to the same thing, this number is (n — A). It 

 will be convenient to term this number X the exponent of 

 irregularity of the permutation, and then {n — X) may be termed 

 the supplement of the exponent of irregularity. The rule in 

 the case of a series of things, all of them different, may con- 

 sequently be stated as follows: " a permutation is positive or 

 negative according as the supplement of the exponent of irre- 

 gularity is even or odd." Consider now a series of things, 

 not all of them different, and suppose that this is derived from 

 the system of the same number of things abc... all of them 

 originally different, by supposing for instance « = 6 = &C., 

 f=g=:Si,c. A given permutation of the system of things not 

 all of them different, is of course derivable under the suppo- 

 sition in question from several different permutations of the 

 series abc... Considering the supplements of the exponents of 

 irregularity of these last-mentioned several permutations, we 

 may consider the given permutation as positive or negative ac- 

 cording as the least of these numbers is even or odd. Hence we 

 obtain the rule, " a permutation of a series of things not all of 

 them different, is positive or negative according as the minimum 

 supplement of irregularity of the permutation is even or odd, 

 the system being considered as a particular case of a system 

 of the same number of things all of them different, and the 

 given permutation being successively considered as derived 

 from the different permutations which upon this supposition 

 reduce themselves to the given permutation." This only 

 differs from the rule, "a permutation of a series of things, not 

 all of them different, is positive or negative according as the 

 minimum number of inversions by which it can be obtained is 

 even or odd, the system being considered, &c.," inasmuch as 

 the former enunciation is based upon and indicates a direct 

 method of determining the minimum number of inversions re- 

 quisite in order to obtain a given permutation; but the latter 

 is, in simple cases, of the easiest application. Asa very simple 



* See on this subject Cauchy's Memoire sur les Arranfjemens, Sec, Exer- 

 cices d' Analyse et de Physique Mathermtique, t. iii, p. 151. 



