12 
THE SYMMETRIC FUNCTION TABLES OF THE FIFTEENTHIC. 
A third formula, based on a system of operators developed by Ham¬ 
mond in The Proceedings of the London Mathematical Society for 1882-83 
(vol. 14), provides a method for calculating any symmetric function of the 
n-thic from the elementary symmetric functions of weight n and lower, l—l 
checks being provided for a coefficient of a term containing l a’ s. The 
operators are unusual in that, while calculation by means of them is very 
simple, they themselves are functions of elementary symmetric functions. 
One application of them to the calculation of symmetric is given by 
Hammond in his paper, his illustration being the calculation of [3 4 1] 
from [3 4 ]. 
A fourth formula in MacMahon’s article provides a check by giving 
the sum of the coefficients of the terms of a given degree in a symmetric 
function, or the sum of the coefficients of a given term in all the symmetric 
functions with a given number of parts in their partitions. He first 
introduces a generalized definition of weight 
1T V —ri/l y _i+ r 2 k v + r 3 A, y+ i+ • • • 
where 
r x =K + }K(K-l)(r-2) 
that is, where r K is the kth of the r-gonal numbers; a definition which, 
it is to be observed, makes w 2 = W. The formula for the sum of the 
coefficients of the terms of the mth degree in the a ’s in the value of 
[xf’xf 2 . . . x* n ~] (or the sum of the coefficients of (xf'xf 2 . . . x„ n ) in all 
of the symmetric functions with m-part partitions) is 
where 
_ )K+m - AK-l)\W m+l 
1 K x \K 2 \...K n \ 
K=K 1 +K 2 +- --+K n 
(17) 
For example, the sum of the coefficients of terms of the fourth degree in 
the a’s in the value of [S^ 1 ! 3 ] is 
( n« +4 -i (5-l) 1(5-1) 
1 ’ 3!1! 1! 
20 
since 
K x = 3 K 2 = 1 K 3 = 0 K, = 0 K 5 = l K 6 = 0 K 7 = 0 . . . Z n = 0 
K = 5 M = 4 TF S = 5g • 1 = 5 
In the following year the same writer published an article in the 
American Journal of Mathematics in which he repeated formula (17) and 
included the additional check that any function except [1 A ] would vanish 
if for each coefficient the reciprocal of the factorial of its suffix were sub¬ 
stituted, (18). Accompanying the article is the thirteenths, arranged like 
Durfee’s twelfthic and checked by the CaylejMBetti law of symmetry, (10), 
as well as by his own checks, (17), (18). 
