$$\sum_{ k\ge 1 } \sum_{ (s_1,...,s_k) } \binom h{ s_1 ,..., s_k }=\binom{m-1}{h-1}$$ where the second summation is taken over all choices of the numbers

$${ s }_{ 1 },...,{ s }_{ k-1 }\ge 0,\quad {s }_{k }\ge 1$$

which satisfy the relations $$\sum _{ i=1 }^{ k }{ { s }_{ i } } =h,\quad \sum _{ i=1 }^{ k }{ i{ s }_{ i } } = m$$