A note on the number of plane partitions and $r$-component multipartitions of $n$

Abstract

Using elementary methods, we prove new formulas for $\pp(n)$, the number of plane partitions of $n$, $\pp_r(n)$, the number
of plane partitions of $n$ with at most $r$ rows, $\pps(n)$, the number of strict plane partitions of $n$
and $\ppso(n)$, the number of symmetric plane partitions of $n$. Also, we give new formulas for $P_r(n)$,
the number of $r$-component multipartitions of $n$.