An extension of carlitz's bipartition identity

Carlitz' bipartition identity is extended to a multipartite partition identity by the introduction of the summatory maximum function: smax(n1, n2,., nr) = n1 + n2 +. + nr (r - 1)min(n1, n2,., nr). Let π0(n1, n2,., nr) denote the number of partitions of (n1, n2,., nr) in which the minimum coordinate of each part is not less then the summatory maximum of the next part. Let πl(nl, n2,., nr) denote the number of partitions of (n1, n2,., nr) in which each part has one of the 2r 1 forms: (a + 1, a, a,.a), (a, a + 1, a,., a). (a, a, a,., a+ 1), (ra + 2, ra + 2,., ra + 2), (ra + 3, ra + 3,., ra + 3),., (ra + r, ra + r,., ra + r). Theorem: π0(n1,., nr) = π1(n1,., nr).