Bijections on pattern avoiding inversion sequences and related objects

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.