# Permutations

Suppose $$S$$ is a finite set having $$n$$ distinct elements. Then a one-one mapping of $$S$$ onto itself is called a permutation of degree $$n$$. The number of elements in the finite set $$S$$ is known as the degree of permutation.

__Symbol for a Permutation__

Let $$S = \left\{ {{a_1},{a_2},{a_3}, \ldots ,{a_n}} \right\}$$ be a finite set having $$n$$ distinct elements. If $$f:S \to S$$ in one-one mapping, then $$f$$ is a permutation of degree $$n$$.

Let $$f\left( {{a_1}} \right) = {b_1}$$, $$f\left( {{a_2}} \right) = {b_2}$$, $$f\left( {{a_3}} \right) = {b_3}$$, …, $$f\left( {{a_n}} \right) = {b_n}$$ where $$\left\{ {{b_1},{b_2},{b_3}, \ldots ,{b_n}} \right\}$$$$ = \left\{ {{a_1},{a_2},{a_3}, \ldots ,{a_n}} \right\}$$ i.e. $${b_1},{b_2},{b_3}, \ldots ,{b_n}$$ is one arrangement of the $$n$$ elements $${a_2}, \ldots ,{a_n}, \ldots $$.

It is customary to write a permutation in a two line symbol. In this notation we write:

\[ f = \left( {\begin{array}{*{20}{c}} {{a_1}}&{{a_2}}&{{a_3}}& \cdots &{{a_n}} \\ {{b_1}}&{{b_2}}&{{b_3}}& \cdots &{{b_n}} \end{array}} \right) \]

i.e. each element in the second row is the $$f – $$ image of the element of the first row lying directly above it.

If $$S = \left\{ {1,2,3,4} \right\}$$ is a finite set of order four then,

\[ f = \left( {\begin{array}{*{20}{c}} 1&2&3&4 \\ 2&4&1&3 \end{array}} \right) \]

\[ g = \left( {\begin{array}{*{20}{c}} 1&2&3&4 \\ 1&3&2&4 \end{array}} \right) \]

\[ h = \left( {\begin{array}{*{20}{c}} 1&2&3&4 \\ 1&2&4&3 \end{array}} \right) \]

etc., are all permutations of degree four. Here in the permutation $$f$$ the elements **1, 2, 3, 4** are replaced respectively by the elements **2, 4, 1, 3. **Thus $$f\left( 1 \right) = 2,f\left( 2 \right) = 4,$$ $$f\left( 3 \right) = 1,f\left( 4 \right) = 3$$. Similarly, $$g\left( 1 \right) = 1,g\left( 2 \right) = 3,$$ $$g\left( 3 \right) = 2,g\left( 4 \right) = 4$$ and $$h\left( 1 \right) = 1,h\left( 2 \right) = 2,$$ $$h\left( 3 \right) = 4,h\left( 4 \right) = 3$$.

Thus for a permutation $$f$$ on $$S$$, we just put the elements of $$S$$ in one row in any order we like and below each element of this row we put down its image under, $$f,g$$ or $$h$$ to obtain another row of elements of $$S$$.