  We study some combinatorial properties of higher-dimensional partitions which
generalize plane partitions. We present a natural bijection between
$d$-dimensional partitions and $d$-dimensional arrays of nonnegative integers.
This bijection has a number of important applications. We introduce a statistic
on $d$-dimensional partitions, called the corner-hook volume, whose generating
function has the formula of MacMahon's conjecture. We obtain multivariable
formulas whose specializations give analogues of various formulas known for
plane partitions. We also introduce higher-dimensional analogues of dual
Grothendieck polynomials which are quasisymmetric functions and whose
specializations enumerate higher-dimensional partitions of a given shape.
Finally, we show probabilistic connections with a directed last passage
percolation model in $\mathbb{Z}^d$.
