Macdonald's plane partition conjecture proposes a formula for the number of cyclically symmetric plane partitions (CSPPs) of a given integer whose Ferrers
diagrams fit inside an box. Macdonald gave a product representation
for the power series whose coefficients were the number of such partitions of .
Let
be the set of all integer points in the first octant such
that a plane partition is defined and . Then is said to be cyclically symmetric if is invariant under the mapping . Let be the number of cyclically symmetric partitions of
such that none of exceed . Let be the box containing all integer points such that , then is the number of cyclically symmetric plane partitions
of
such that . Now, let be the set of all the orbits in . Finally, for each point in , let its height
(1)
and for each in , let be the number of points in (either 1 or 3) and write
(2)
Then Macdonald conjectured that
(3)
(4)
(5)
(Mills et al. 1982, Macdonald 1995), where the latter form is due to Andrews
(1979).
The first few polynomials are
(6)
(7)
(8)
(9)
which converge to the polynomial with coefficients 1, 1, 0, 0, 1, 0, 0, 2, 1, 0,
2, 1, 0, 4, 3, 0, 5, 4, 0, 8, 8, ... (OEIS A096419).
Andrews (1979) proved the case, giving the total number of CSPPs fitting inside an
box. The general case was proved by Mills et al. (1982).
Andrews, G. E. "Plane Partitions (III): The Weak Macdonald Conjecture." Invent. Math.53, 193-225, 1979.Andrews,
G. E. "Macdonald's Conjecture and Descending Plane Partitions." In
Combinatorics,
Representation Theory and Statistical Methods in Groups (Ed. T. V. Narayana,
R. M. Mathsen, and J. G. Williams). New York: Dekker, pp. 91-106,
1980.Bressoud, D. Proofs
and Confirmations: The Story of the Alternating Sign Matrix Conjecture. Cambridge,
England: Cambridge University Press, 1999.Bressoud, D. and Propp, J.
"How the Alternating Sign Matrix Conjecture was Solved." Not. Amer.
Math. Soc.46, 637-646.Macdonald, I. G. "Some conjectures
for Root Systems." SIAM J. Math. Anal.13, 988-1007, 1982.Macdonald,
I. G. Symmetric
Functions and Hall Polynomials, 2nd ed. Oxford, England: Oxford University
Press, 1995.Mills, W. H.; Robbins, D. P.; and Rumsey, H. Jr.
"Proof of the Macdonald Conjecture." Invent. Math.66, 73-87,
1982.Morris, W. G. Constant Term Identities for Finite and Affine
Root Systems: Conjectures and Theorems. Ph.D. thesis. Madison, WI: University
of Wisconsin, 1982.Sloane, N. J. A. Sequence A096419
in "The On-Line Encyclopedia of Integer Sequences."
box. Macdonald gave a product representation
for the power series whose coefficients 
were the number of such partitions of 
.
be the set of all integer points 
in the first octant such
that a plane partition 
is defined and 
. Then 
is said to be cyclically symmetric if 
is invariant under the mapping 
. Let 
be the number of cyclically symmetric partitions of 
such that none of 
exceed 
. Let 
be the box containing all integer points 
such that 
, then 
is the number of cyclically symmetric plane partitions
of 
such that 
. Now, let 
be the set of all the orbits in 
. Finally, for each point 
in 
, let its height
in 
, let 
be the number of points in 
(either 1 or 3) and write
![product_(i=1)^(m)[(1-q^(3i-1))/(1-q^(3i-2))product_(j=i)^(m)(1-q^(3(m+i+j-1)))/(1-q^(3(2i+j-1)))],](/images/equations/MacdonaldsPlanePartitionConjecture/Inline37.svg)
case, giving the total number of CSPPs fitting inside an
box. The general case was proved by Mills et al. (1982).
