Vol. 72, No. 2, 1977

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

George E. Andrews

Vol. 72 (1977), No. 2, 283–291

A plane partition of the integer n is a representation of n in the form n = i,j1nij where the integers nij are nonnegative and nij ni,j+1, nij ni+1,j. In 1898, MacMahon conjectured that the generating function for the number of symmetric plane partitions (i.e., nij = nji) with each part at most m (i.e., n11 m) and at most s rows (i.e., nij = 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., nij > ni,j+1 whenever nij > 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
Received: 1 March 1976
Published: 1 October 1977
George E. Andrews
Department of Mathematics
The Pennsylvania State University
109 McAllister Building
University Park PA 16802-7000
United States