= Partition into distinct parts
= Partitions into distinct parts
{synonym}
An <integer partition> with no repeated part. For example, the <partitions into distinct parts> of $5$ are $5$, $4+1$, and $3+2$. If $d(i,n)$ counts those with all parts at most $i$, separating those that omit $i$ from those that use it exactly once gives
$$
d(i,n)=d(i-1,n)+d(i-1,n-i)
$$
for $i,n\geq1$, with $d(i,0)=1$, $d(0,n)=0$ for $n>0$, and negative remaining sums counted as zero. Removing the unique part $i$ gives a <bijection> proving the second term; retaining the same maximum $i$ there would instead allow repetitions.
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