Product of some binomial coefficients

Question

Is there a special name of the following products of binomial coefficients? If there is, is there a good document listing some properties of these?

$$\prod_{i=1}^{k-1}{n_i-1\choose n_{i+1}}={n_1-1 \choose n_2}{n_2-1\choose n_3}...{n_{k-1}-1 \choose n_{k}}$$

Or, what are possible enumerative interpretations of this formula?

Answer 1

Using $$ \binom{n_i-1}{n_{i+1}}=\frac{n_i-n_{i+1}}{n_{i}}\binom{n_i}{n_{i+1}}, $$ your product can be written in terms of a multinomial coefficient: $$ \frac{(n_1-n_2)(n_2-n_3)\cdots(n_{k-1}-n_k)}{n_1n_2\cdots n_{k-1}}\binom{n_1}{n_1-n_2,n_2-n_3,\dots, n_{k-1}-n_k,n_k} $$