TITLE: Entropy of chain decompositions of posets
QUESTION [7 upvotes]: 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.

REPLY [3 votes]: The answer is no. Take a chain $a_1>\dots>a_{100}$ and add two elements $b>a_{51}$ and $c<a_{50}$ (with no relations not implied by these ones). The maximum of $f(\mathcal C)$ is achieved on the decomposition $\{a_1,\dots,a_{100}\}\sqcup\{b\}\sqcup\{c\}$, while the maximal antichain has cardinality 2 (due to the decomposition into two chains $\{a_1,\dots,a_{50},c\}\sqcup \{b,a_{51},\dots,a_{100}\}$).