Plane Partition

n_{i,j} \ge n_{i,j+1} \quad\mbox{and}\quad n_{i,j} \ge n_{i+1,j} \, .

Define the sum of the plane partition by
:

n=\sum_{i,j} n_{i,j} \,

and let PL(''n'') denote the number of plane partitions with sum ''n''.

For example, there are six plane partitions with sum 3:
:

\begin{matrix} 1 & 1 & 1 \end{matrix} 
\qquad \begin{matrix} 1 & 1 \ 1 & \end{matrix}
\qquad \begin{matrix} 1 \ 1 \ 1 & \end{matrix}
\qquad \begin{matrix} 2 & 1 & \end{matrix}
\qquad \begin{matrix} 2 \ 1 & \end{matrix}
\qquad \begin{matrix} 3 \end{matrix}

so PL(3) = 6.

The Generating Function for PL(''n'') of planar partitions of n is
:

\sum_{n=0}^{\infty} \mbox{PL}(n) \, x^n = 1+x+3x^2+6x^3+13x^4+24x^5+\cdots.

EXTERNAL LINKS

	CATEGORIES ABOUT PLANE PARTITION

	combinatorics

	discrete mathematics

Information About