**Theorem[Bender-Knuth]:** There is a one-to-one correspondence between plane partitions of $N$, on the one hand, and infinite matrices $a_{ij}$ ($i,j\geq 1$) of non-negative integer entries which satisfy

on the other.

**Exercise:** Construct a bjiection between the above matrices and plane partitions of $N$. See page 83 upwards of the book, Combinatorial Algorithms by Nijenhuis and Wilf to see details of how the bijection works.

The natural generalization for a matrix analog of solid partitions would be the following. Consider the infinite hypermatrices $a_{ijk}$ ($i,j,k\geq 1$) of non-negative integer entries which satisfy

(2)**Claim:** The number of hypermatrices is generated by MacMahon's formula i.e., the one that gave the *wrong* answer for solid partitions.

**Exercise:** Attempt to construct a bjiection between the above matrices and solid partitions of $N$. See if it fails precisely where the MacMahon formula fails i.e., at $N=6$.