Powers of adjacency matrix doesn't seem to correspond to observed number of paths on graph
Solution 1:
The powers of the adjacency matrix count the number of $i \to j$ walks, not paths. A walk can repeat vertices, while a path cannot.
The walks are:
$1 \to 3 \to 4 \to 2$
$1 \to 3 \to 1 \to 2$
$1 \to 2 \to 1 \to 2$
$1 \to 2 \to 4 \to 2$
$1 \to 4 \to 1 \to 2$