JMO 1998 Problem 4

Let $c_{n, m}$ be the number of permutations of $\{1, \ldots, n\}$ which can be written as the product of $m$ transpositions of the form $(i, i+1)$ for some $i=1, \ldots, n-1$ but not of $m-1$ suct transpositions. Prove that for all $n \in \mathbb{N}$, $$\sum_{m=0}^\infty c_{n, m}t^{m}= \prod_{i=1}^{n}(1+t+\cdots+t^{i-1}).$$