A structure theorem for posets admitting a “strong” chain partition: A generalization of a conjecture of Daykin and Daykin (with connections to probability correlation inequalities)
- Farley, Jonathan David
DOIAbstract
AbstractSuppose a finite poset P is partitioned into three non-empty chains so that, whenever p, q∈P lie in distinct chains and p<q, then every other element of P is either above p or below q.In 1985, the following conjecture was made by David Daykin and Jacqueline Daykin: such a poset may be decomposed into an ordinal sum of posets ⊕i=1nRi such that, for 1⩽i⩽n, one of the following occurs:(1)Ri is disjoint from one of the chains of the partition; or(2)if p, q∈Ri are in distinct chains, then they are incomparable.The conjecture is related to a question of R. L. Graham's concerning probability correlation inequalities for linear extensions of finite posets.In 1996, a proof of the Daykin–Daykin conjecture was announced (by two other mathemati...
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