Macdonald's Plane Partition Conjecture

Author

Eric W. Weisstein
February 10, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/MacdonaldsPlanePartitionConjecture.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/MacdonaldsPlanePartitionConjecture.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Sum

SumM[m_]:=Product[((1-(q^(3i-1)))/(1-(q^(3i-2))))Product[((1-(q^(3(m+i+j-1))))/(1-(q^(3(2i+j-1))))),{j,i,m}],{i,m}]

Expand/@Factor/@SumM/@Range[0,3]//ColumnForm

CoefficientList[#,q]&/@Factor/@SumM/@Range[0,5]//ColumnForm

Series[SumM[100],{q,0,20}]

CoefficientList[%,q]

Product[((1-(q^(3(m+i+j-1))))/(1-(q^(3(2i+j-1))))),{j,i,m}]//FullSimplify

Product[((1-(q^(3i-1)))/(1-(q^(3i-2))))%,{i,m}]

FullSimplify[%,(m|i)∈Integers&&i>0&&m>0]