  A partition into distinct parts is refinable if one of its parts $a$ can be
replaced by two different integers which do not belong to the partition and
whose sum is $a$, and it is unrefinable otherwise. Clearly, the condition of
being unrefinable imposes on the partition a non-trivial limitation on the size
of the largest part and on the possible distributions of the parts. We prove a
$O(n^{1/2})$-upper bound for the largest part in an unrefinable partition of
$n$, and we call maximal those which reach the bound. We show a complete
classification of maximal unrefinable partitions for triangular numbers,
proving that if $n$ is even there exists only one maximal unrefinable partition
of $n(n+1)/2$, and that if $n$ is odd the number of such partitions equals the
number of partitions of $\lceil n/2\rceil$ into distinct parts. In the second
case, an explicit bijection is provided.
