TY - JOUR
T1 - Plane partitions (II)
T2 - The equivalence of the bender-knuth and macmahon conjectures
AU - Andrews, George E.
PY - 1977/10
Y1 - 1977/10
N2 - 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.
AB - 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.
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U2 - 10.2140/pjm.1977.72.283
DO - 10.2140/pjm.1977.72.283
M3 - Article
AN - SCOPUS:84972496677
SN - 0030-8730
VL - 72
SP - 283
EP - 291
JO - Pacific Journal of Mathematics
JF - Pacific Journal of Mathematics
IS - 2
ER -