Permutations

Shuffling

$(a,b,c)$

$(a,c,b)$

$(b,a,c)$

$(b,c,a)$

$(c,a,b)$

$(c,b,a)$

Actions

$(0, 1, 2) \rightarrow (0,1,2)$

$(0, 1, 2) \rightarrow (0,2,1)$

$(0, 1, 2) \rightarrow (1,0,2)$

$(0, 1, 2) \rightarrow (1,2,0)$

$(0, 1, 2) \rightarrow (2,0,1)$

$(0, 1, 2) \rightarrow (2,1,0)$

Positional Notation

Apply $(1,0,2)$ to $(a,b,c)$ giving $(b,a,c)$.

Sometimes written as

$\begin{pmatrix} 0 & 1 & 2 \\ 1 & 0 & 2 \end{pmatrix}$

Combining Permutations

Apply $(1, 0, 2)(2, 1, 0)$ to $(a,b,c)$.

First, apply $(2,1,0)$ to $(a,b,c)$ giving $(c,b,a)$.

Then apply $(1,0,2)$ to $(c,b,a)$ giving $(b,c,a)$

Note that we can apply $(1,0,2)$ to $(2,1,0)$ giving $(1,2,0)$.

See $(1,2,0)$ applied to $(a,b,c)$ also gives $(b,c,a)$.