Let $\mathcal{X}$ be a finite set. Let $\mathcal{A} = \{A_1,\dots,A_N\}$, where $N\le |\mathcal{X}|$, be a partition of $\mathcal{X}$. Define $$\small H(\mathcal{A}) = 1+\frac{1}{2}+\dots + \frac{1}{|A_1|} + \dots + 1 + \frac{1}{2}+\dots + \frac{1}{|A_N|}.$$
Since $1+\frac{1}{2}+\dots + \frac{1}{|A_i|}\le \log |A_i| +1$, one obvious upper bound for $H(\mathcal{A})$ is $\log(|A_1|\dots |A_N|) + N \le N \log\Big(\frac{|\mathcal{X}|}{N}\Big) + N$ by the AM-GM inequality.
Does anyone aware of any other upper bound on $H(\mathcal{A})$? Is this sum studied before? Any help is greatly appreciated.