Elementary Proof of MacMahon's Conjecture
David M. Bressoud
DOI: 10.1023/A:1008667105627
Abstract
Major Percy A. MacMahon”s first paper on plane partitions [4] included a conjectured generating function for symmetric plane partitions. This conjecture was proven almost simultaneously by George Andrews and Ian Macdonald, Andrews using the machinery of basic hypergeometric series [1] and Macdonald employing his knowledge of symmetric functions [3]. The purpose of this paper is to simplify Macdonald”s proof by providing a direct, inductive proof of his formula which expresses the sum of Schur functions whose partitions fit inside a rectangular box as a ratio of determinants.
Pages: 253–257
Keywords: plane partition; symmetric plane partition; Schur function
Full Text: PDF
References
1. George Andrews, “Plane partitions (I): The MacMahon conjecture,” Studies in Foundations and Combinatorics, Advances in Mathematics Supplementary Studies 1 (1978), 131-150.
2. Jacques Désarménien, “Une generalisation des formules de Gordon et de MacMahon,” C.R. Acad. Sci. Paris Series I, Math. 309(6) (1989), 269-272.
3. I.G. Macdonald, Symmetric Functions and Hall Polynomials, second edition, Oxford University Press, 1995.
4. P.A. MacMahon, “Partitions of numbers whose graphs possess symmetry,” Trans. Cambridge Phil. Soc. 17 (1898-1899), 149-170.
2. Jacques Désarménien, “Une generalisation des formules de Gordon et de MacMahon,” C.R. Acad. Sci. Paris Series I, Math. 309(6) (1989), 269-272.
3. I.G. Macdonald, Symmetric Functions and Hall Polynomials, second edition, Oxford University Press, 1995.
4. P.A. MacMahon, “Partitions of numbers whose graphs possess symmetry,” Trans. Cambridge Phil. Soc. 17 (1898-1899), 149-170.