**MSC:**- 06A07 Combinatorics of partially ordered sets
- 05D05 Extremal set theory

by the subword relation. It is known that $SO(2)$ falls into the class of

Macaulay posets, i.e. there is a theorem of Kruskal--Katona type for $SO(2)$.

As the corresponding linear ordering of the elements of $SO(2)$ the

\emph{vip}--order can be chosen.

Daykin introduced the \emph{V}--order which generalizes the \emph{vip}--order

to the $n\ge 2$ case. He conjectured that the \emph{V}--order gives a

Kruskal--Katona type theorem for $SO(n)$.

We show that this conjecture fails for all $n\ge 3$ by explicitely giving a

counterexample. Based on this, we prove that for no $n\ge 3$ the subword order

$SO(n)$ is a Macaulay poset.