16 
[i°4 
104. 
ON THE THEORY OF PERMUTANTS. 
[From the Cambridge and Dublin Mathematical Journal, voi. vii. (1852), pp. 40—51.] 
A form may by considered as composed of blanks which are to be filled up by 
inserting in them specializing characters, and a form the blanks of which are so filled 
up becomes a symbol. We may for brevity speak of the blanks of a symbol in the 
sense of the blanks of the form from which such symbol is derived. Suppose the 
characters are 1, 2, 3, 4, ..., the symbol may always be represented in the first 
instance and without reference to the nature of the form, by V 1234 ... And it will be 
proper to consider the blanks as having an invariable order to which reference will 
implicitly be made; thus, in speaking of the characters 2, 1, 3, 4,... instead of as 
before 1, 2, 4,... the symbol will be V 2134 ... instead of V 1234 .... When the form is 
given we shall have an equation such as 
F"1234 = Pi 2 Qi P4 • • • Or = 1 12 1 3 4 • • • &C., 
according to the particular nature of the form. 
Consider now the characters 1, 2, 3, 4, ... , and let the primitive arrangement and 
every arrangement derivable from it by means of an even number of inversions or 
interchanges of two characters be considered as positive, and the arrangements derived 
from the primitive arrangement by an odd number of inversions or interchanges of 
two characters be considered as negative ; a rule which may be termed “ the rule of 
signs.” The aggregate of the symbols which correspond to every possible arrangement 
of the characters, giving to each symbol the sign of the arrangement, may be termed 
a “ Permutant ; ” or, in distinction from the more general functions which will presently 
be considered, a simple permutant, and may be represented by enclosing the symbol 
in brackets, thus ( F 1234 ...). And by using an expression still more elliptical than the 
blanks of a symbol, we may speak of the blanks of a permutant, or the characters 
of a permutant.