<s>[INST] Problem involving permutation matrices from Michael Artin's book. Let $p$ be the permutation $(3 4 2 1)$ of the four indices.  The permutation matrix associated with it is 
$$ P =
\begin{bmatrix}
0 & 0 & 1 & 0 \\ 
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0
\end{bmatrix}
$$
This is the matrix that permutes the components of a column vector.
The problems asks you to decompose $p$ into transpositions and show that the associated matrix product equals the above matrix.  However, I'm not getting that it does and I've run through it several times.  Here are my calculations:
$p = (12)(14)(13)$
$$
P_{(12)} = \begin{bmatrix}
0 & 1 & 0 & 0 \\ 
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}, 
P_{(14)} =\begin{bmatrix}
0 & 0 & 0 & 1 \\ 
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0
\end{bmatrix},
P_{(13)}=\begin{bmatrix}
0 & 0 & 1 & 0 \\ 
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
$$
However, 
$$P_{(12)} (P_{(14)} P_{(13)}) = 
\begin{bmatrix}
0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0
\end{bmatrix}
\neq P$$
I don't see where I've made a mistake.
 [/INST] The permutation is indeed $(1342)$, but this decomposes as $(13)(14)(12)$, so you should compute
$$
P_{(13)} P_{(14)} P_{(12)}
=
\begin{bmatrix}
0 & 0 & 1 & 0 \\ 
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\cdot
\begin{bmatrix}
0 & 0 & 0 & 1 \\ 
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0
\end{bmatrix}
\cdot
\begin{bmatrix}
0 & 1 & 0 & 0 \\ 
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
=
P.
$$
You should compose both permutations and matrices consistently left-to-right.
</s>