We consider the poset
SO(
n) of all words over an
n-element alphabet ordered 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
vip-order can be chosen.
Daykin introduced the
V-order which generalizes the
vip-order to the
n
2 case. He conjectured that the
V-order gives a Kruskal–Katona type theorem for
SO(
n).
We show that this conjecture fails for all
n3 by explicitly giving a counterexample. Based on this, we prove that for no
n3 the subword order
SO(
n) is a Macaulay poset.
Mathematics Subject Classification (2000): 06A07 - 05D05 - 68R15