Plane partitions (II): The equivalence of the bender-knuth and macmahon conjectures

A plane partition of the integer n is a representation of n in the form n = Σi, i≥1 n_ij where the integers n_ij are nonnegative and n_ij ≧ n_{i, j+l}n_ij ≧ n_i+1j. In 1898, MacMahon conjectured that the generating function for the number of symmetric plane partitions (i.e., n_ij = n_ji) with each part at most m (i.e., n₁₁≦m) and at most s rows (i.e., n_ij = 0 for i> s) has a simple closed form. In 1972, Bender and Knuth conjectured that a simple closed form also exists for the generating function for plane partitions having at most s rows, n₁₁ ≦ m and strict decrease along rows (i.e., n_ij > n_{i, j+1} whenever n_ij > 0). The main theorem of this paper establishes that each conjecture follows immediately from the other. In the first paper of this series, MacMahon’s conjecture was proved. Hence a corollary of the main theorem here is the truth of the Bender-Knuth conjecture; the Bender-Knuth conjecture has also been proved in a different manner by Basil Gordon.