These are the principal properties of symmetric functions that will be of use.
9. If we take any assemblage of letters such as and are not
concerned with the order in which these letters are written, we have a
‘Combination’ of the letters. If however the order in which the letters
are written be taken into account, we have a ‘Permutation’ of the
letters. In the present case we have twelve permutations, viz.
10. In a similar manner if we take any collection of integers which
add up to a given integer we have as above defined (Art. 3) a partition
of the given number; here no account is taken of the order in which
the parts of the partition may be written; but if order has to be taken
into account each way of writing the parts is called a ‘Composition’ of
the number, such composition appertaining to the particular partition
which is involved. Thus of the number , is a partition
which gives rise to the twelve compositions:
and it will be noticed that the compositions which appertain to the
partition of the number are in correspondence with the permutations
of the combination .
Moreover, in general the compositions which appertain to any given partition of a number are in correspondence with the permutations of a certain combination of letters.
11. In pursuing the main object of this book, namely the study of
the algebra of symmetric functions together with those theories of
combination, permutation, arrangement, order and distribution which are
summed up in the title ‘Combinatory Analysis,’ it is important to have
some specific rules for arranging the order in which the terms of algebraic
expressions are written down.