Let $P$ be a finite poset and let $\,\mathcal{C}=(C_1,\ldots,C_\ell)\,$ be its decomposition into chains. We can define $$ f(\mathcal{C}) = |C_1|! \, \cdots \,|C_\ell|! $$ and ask for what $\mathcal{C}$ we have $f(\mathcal{C})$ maximal. Roughly speaking, the smaller is $\ell$ and the less evenly distributed are $|C_i|$, the better. Specifically, I am interested in the following.

**Question:** *Does $f(\mathcal{C})$ maximize for the number $\ell$ of chains equal to the size of the maximal antichain in $P$?*

**Motivation:** If true, this would be an unusual extension of Dilworth's theorem. I am really interested in bounds on the number $e(P)$ of linear extensions, and it is easy to see that $e(P)\le |P|!/f(\mathcal{C})$ for all chain decompositions $\mathcal{C}$. The connection to entopy is also standard: take $\log e(P)$ in the last equation and think of $|C_i|/|P|$ as a probability distribution.

**Refs disclaimer:** There is a large number of papers bounding $e(P)$ in terms of the entropy, see e.g. here, here and here. They are not completely unrelated, but do not seem concerned with this type of questions.