476 
Proceedings of Royal Society of Edinburgh. [sess. 
CONTENTS OF THIS PAPER. 
Introduction : — 
§ 1. Cramer’s rule of signs : “ inverted-pairs.” 
§ 2. Rothe’s rule : “ interchanges.” 
§ 3. Cauchy’s rule : “ circular substitutions.” 
§ 4. Drink water’s rule : “ moves.” 
§5. Jenkins’ rule : “ even circular substitutions.” 
§ 6. Summary regarding the rules. 
Inverted-pairs : — 
§ 7. Grouping of permutations according to number, S, of inverted-pairs in 
each. 
§ 8. Modes of fully specifying S for any permutation. 
§ 9. Converse problem of obtaining permutation from detailed specifica- 
tion of 
§ 10. Tabulation of the permutations and corresponding values of 5 for the 
case of 4 elements. 
§ 11. Study of the law of development of the values of 3 in the table. 
§ 12. Problem of determining the ordinal number of a permutation from 
the detailed form of and the converse problem. 
§ 13. Values of £ for case of n elements, deducible from those for case of 
n- 1 elements. 
§ 14. Question of the number, Yn,S, of permutations having £ inverted- 
pairs : theorem Yn,8 = Yn,%n(n 1) 8- 
§ 15. Difference- equation of Yn,S. 
§ 16. Table of values of Yn,8 from n=l to n = ll. 
§ 17. Shorter form of difference-equation. 
§ 18. Second method of constructing table of values of Yn,S. 
§ 19. Relation of this method to theorem of § 7. 
§ 20. Generating function of Yn,S : expression of Yn,8 in terms of combina- 
torial. 
§ 21. Case of Yn,S where t=n. 
§ 22. Cases of Yn,S where 5 $>n r 
§ 23. Connection of Y n ,S with a problem in combinations : theorem 
Vw,S = Cl+2+3+. . .+(w-l),8. 
§ 24. Explanation of this connection : consequent deduction of theorem 
of § 14. 
§ 25. Connection of V n> 8 with a problem in the factorisation of integers of 
a certain form. 
§ 26. Further elucidation of the connection of the three problems. 
§ 27. Theorem regarding value of 5 for conjugate permutation. 
