  The number of inversion sequences avoiding two patterns $101$ and $102$ is
known to be the same as the number of permutations avoiding three patterns
$2341$, $2431$, and $3241$. This sequence also counts the number of
Schr\"{o}der paths without triple descents, restricted bicolored Dyck paths,
$(101,021)$-avoiding inversion sequences, and weighted ordered trees. We
provide bijections to integrate them together by introducing $F$-paths.
Moreover, we define three kinds of statistics for each of the objects and count
the number of each object with respect to these statistics. We also discuss
direct sums of each object.
