Nonexistence of a Kruskal-Katona Type Theorem for Subword Orders

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 n≥3 by explicitly giving a counterexample. Based on this, we prove that for no n≥3 the subword order SO(n) is a Macaulay poset.

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Correspondence to Uwe Leck.

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Leck, U. Nonexistence of a Kruskal-Katona Type Theorem for Subword Orders. Combinatorica 24, 305–312 (2004).

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Mathematics Subject Classification (2000):

  • 06A07
  • 05D05
  • 68R15