We give an alternate proof of one of the inequalities proved recently for martingales (= sums of martingale differences) in a non-commutative $L_{p}$-space, with $1 \lt p \lt \infty$, by Q. Xu and the author. This new approach is restricted to $p$ an even integer, but it yields a constant which is $O(p)$ when $p \rightarrow \infty$ and it applies to a much more general kind of sum which we call $p$-orthogonal. We use mainly combinatorial tools, namely the Möbius inversion formula for the lattice of partitions of a $p$-element set.