Given a Vector field $X$ on a manifold $M$, there is a Linear map

$i_X : \Omega^p (M) \to \Omega^{p-1} (M)$

called the interior product, such that

• $i_X df = X(f)$
• $i_X(\alpha \wedge \beta) = i_X \alpha \wedge \beta + (-1)^p \alpha \wedge i_X \beta$ ir $\alpha \in \Omega^p (M)$

This already defines the product, and it can be written in coordinates as:

$i_V (\alpha) =k \sum\limits_{1 \leq i_1 ,...,i_k \leq n} v^{i_1} \alpha_{i_1 i_2 ... i_k} dx_{i_2} \wedge dx_{i_3} \wedge ... \wedge dx_{i_k}$

That factor of $k$ wouldn't be there if we had adopted the convention of increasing indices. But because we are summing over all indices it's there (see here for instance).

See here