I would like to know if I have discovered or merely rediscovered the following pretty fact.

A partition of $[0,1]$ into intervals of lengths $p_{i, i=1\ldots n}$ induces a probability distribution with entropy $-\sum p_i \log_2 p_i$; call this also the entropy of the partition.

For a given $n$, entropy gets maximized by the uniform partition, namely taking all the $p_i$ equal to $1/n$.

Alternatively, one can generate a type of random partition by randomly sampling $n-1$ points according the uniform distribution on $[0,1]$, and then making the sampled points the endpoints of intervals.

The pretty fact: as $n$ grows large, the average amount by which the entropy of the uniform partition exceeds the entropy of this sort of random partition tends to a limit with the simple expression $(1-\gamma)/\ln(2)=0.609948863612\ldots$.

**Question**: does this fact appear anywhere in the literature? (Also interested if it's a folk theorem.)