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

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Abstract

A plane partition of the integer n is a representation of n in the form n = Σi, i≥1 nij where the integers nij are nonnegative and nij ≧ ni, j+lnij ≧ ni+1j. 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 conjecture has also been proved in a different manner by Basil Gordon.

Original languageEnglish (US)
Pages (from-to)283-291
Number of pages9
JournalPacific Journal of Mathematics
Volume72
Issue number2
DOIs
StatePublished - Oct 1977

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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