  We prove combinatorially some identities related to Euler's partition
identity (the number of partitions of $n$ into distinct parts equals the number
of partitions of $n$ into odd parts). They were conjectured by Beck and proved
by Andrews via generating functions.
  Let $a(n)$ be the number of partitions of $n$ such that the set of even parts
has exactly one element, $b(n)$ be the difference between the number of parts
in all odd partitions of $n$ and the number of parts in all distinct partitions
of $n$, and $c(n)$ be the number of partitions of $n$ in which exactly one part
is repeated. Then, $a(n)=b(n)=c(n)$. The identity $a(n)=c(n)$ was proved
combinatorially (in greater generality) by Fu and Tang. We prove
combinatorially that $a(n)=b(n)$ and $b(n)=c(n)$. Our proof relies on
bijections between a set and a multiset, where the partitions in the multiset
are decorated with bit strings. Let $c_1(n)$ be the number of partitions of $n$
such that there is exactly one part occurring three times while all other parts
occur only once and let $b_1(n)$ to be the difference between the total number
of parts in the partitions of $n$ into distinct parts and the total number of
different parts in the partitions of $n$ into odd parts. We prove
combinatorially that $c_1(n)=b_1(n)$. In addition to these results by Andrews,
we prove combinatorially that $b_1(n)=a_1(n)$, where $a_1(n)$ counts partitions
of $n$ such that the set of even parts has exactly one element and satisfying
some additional conditions. We also treat the case when exactly one part occurs
twice while all other parts occur only once.
