WHICH SATISFY GIVEN CONDITIONS. 
143 
is even or odd; thus in the expression for [11-111], the term (14)(13)(12)(11) has foul- 
factors, and is therefore + , the term (113)(12)(11) has three factors, and is therefore — . 
The numerical coefficients are obtained as follows. There is a common factor derived 
from the expression in [ ] on the left-hand side of the equation; viz. for [11111], which 
contains five equal symbols, this factor is 1.2.3.4.5, =120; for [1112], which con- 
tains three equal symbols, it is 1.2.3, =6; and so on (for a symbol such as [11222] 
containing two equal symbols, and three equal symbols, the factor would be 1.2. 1.2. 3, 
=12, and so in other similar cases). In any term on the right-hand side of the equa- 
tion, we must for a factor such as (11), which contains two equal symbols, multiply 
by for a factor such as (111), which contains three equal symbols, multiply by 
and so on. And in the case where a term (as, for example, the term (122)(1 1) or 
(23)(12)(11), vide supra) occurs more than once, the term is to be taken account of each 
time that it occurs ; or, what is the same thing, since the coefficient obtained as above is 
the same for each occurrence, the coefficient obtained as above is to be multiplied by the 
number of the occurrences of the term. For example, taking in order the several terms 
of the expression for [1112], the common factor is =6, and the several coefficients are 
6, 6.1, 6.1, 6.1x2, 6.1, 6.1, 6. 1-1x2, 6.1; 
and similarly in the expression for [11111] the common factor is 120, and the coeffi- 
cients taken in order are 
120.1, 120.1, 120.1.1.1, &c„ 
without there being in this case any coefficient with a factor arising from the plural 
occurrence of the term. 
The foregoing result was established by induction, and I have not attempted a general 
proof. 
I observe by way of a convenient numerical verification, that in each equation the 
sum of the coefficients (taken with their proper signs) is (— ) M_1 1.2 . . (n — 1); if n be 
the number of parts in the [ ] (n= 5 for [11111], =4 for [1112] &c.), and moreover, 
that the sum of these sums each multiplied by the proper polynomial coefficient and the 
whole increased by unity is =0 ; viz. for 
[14] [23] [113] [122] [1112] [11111], 
the sums of the coefficients are 
— 1, —1, +2, +2, —6, +24 respectively, 
and we have 
l+5(— 1)+10(— 1)+10(2) + 15(2)+10(— 6)+l(24), =75-75, =0. 
If we have any five distinct things (a, b, c, d, e), then the polynomial coefficients 5, 10, 
10, 15, 10, 1 denote respectively the number of ways in which these can be partitioned 
in the forms 14, 23, 113, 122, 1112, 11111 respectively, and the last-mentioned theorem 
is thus a theorem in the Partition of Numbers. 
