Plane partitions. II. The equivalence of the Bender-Knuth and MacMahon conjectures

George E. Andrews

Vol. 72 (1977), No. 2, 283–291

DOI: 10.2140/pjm.1977.72.283

Abstract

A plane partition of the integer n is a representation of n in the form n = ∑ _i,j≧1n_ij where the integers n_ij are nonnegative and n_ij ≧ n_i,j+1, n_ij ≧ n_i+1,j. 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 coniecture has also been proved in a different manner by Basil Gordon.

Mathematical Subject Classification 2000

Primary: 05A17

Milestones

Received: 1 March 1976

Published: 1 October 1977