A combinatorial proof of the Chan--Robbins--Yuen volume formula

The Chan–Robbins–Yuen polytope ${\mathrm{C}\mathrm{R}\mathrm{Y}}_{m+1}$ has normalized volume ${\prod }_{j=1}^{m-1}{C}_j$, a product of Catalan numbers. Zeilberger proved the formula using the Morris constant-term identity; finding a combinatorial proof remained an open problem. We give such a proof by counting nonnegative integer flows. We add an extra edge from each vertex to a common sink and count flows according to how many of these edges carry positive flow. A double count proves a special case of a recurrence of Morales and Shi and gives these counts in terms of Narayana numbers. Deleting the first vertex then yields the Catalan product. We also give an explicit bijection for the recurrence and prove the corresponding reduction formula posed by Postnikov.