Given two permutations $p_1$ and $p_2$ in factorial notation, is there a direct algorithm which computes their composition directly, i.e. without translating to a different notation or via computing the action on a sample vector?
For discreteness, if you could show how $(2\ 2\ 0\ 1\ 0)$ composed with $(3\ 3\ 0\ 0\ 0)$ gives $(4\ 1\ 1\ 1\ 0)$, that would help a lot. Note that I explicitly show the trailing $0$ in the above permutations (even though this is usually omitted). The reason for that is that I am working on some codes in Agda with a dependently-typed representation of permutations (in factorial notation), and Agda forces me to be pedantically precise.