340 PROFESSOR TAIT ON KNOTS. 



APPENDIX 



Note on a Problem in Partitions. By Professor Tait. 



(Read July 7, 1884.) 



In the partition method of constructing knots of any order, n, of knottiness, we have to select from 

 the group of partitions of 2n those only in which no part is greater than n, and no part less than 2. 



Thus, as given in the text, § 6, we have for sevenfold knottiness the series of partitions of 14; — 

 but they are now arranged below in classes according to the value of tbe largest partition. 



77 



662 



554 



4442 



33332 



2222222 



752 



653 



5522 



4433 



332222 





743 



644 



5432 



44222 







7322 



6422 



5333 



43322 









6332 



53222 



422222 









62222 











It is an interesting inquiry to find how many there are in each class, for any value of n. The number 

 of classes is obviously n - 1 ; and, if we remove from each the first partition (i.e., that which is not in- 

 ferior to any of the others), the remainders form a new set of classes of partitions which we may desig- 

 nate as 



Pn > Pu+1 1 Pn+2 ) • • ■ P'ln-1 



respectively; — where p\ is defined as the number of partitions of s, in which no partition is greater than 

 r, and none less than 2. 



Without explicitly introducing finite differences or generating functions it is easy to calculate the 

 values of the quantity p\; — and to put them in a table of double entry which can be developed to any 

 desired extent by the simplest arithmetical processes. The method is similar to one which I employed 

 some years ago for the solution of a problem in Arrangements (Proc. R.S.U., viii. 37, 1872). 



In the first place we see at once that if r>s 



V\ =p\ ■ 



Thus, if r denote the column, and s the row, of the table in which p T , occurs, all numbers in the row 

 following p\ are equal to it. Thus the values of p\ enable us to fill up half the table. In the remain- 

 ing half r is less than s ; and by a dissection of this class of partitions, similar to that which was given 

 above, we see that 



l>: =l?.- r +P r .:ln+ • • ■ ■+P i ,-2+pl-i-^P°., 



where the two last terms obviously vanish ; and the first term is obviously 1 in the case of r = s, uni 

 r<2, when it vanishes. 



