Journal of Combinatorial Theory Series A
Towards bounding sequential patterns
Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining
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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.