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A correlational inequality for linear extensions of a poset. (English) Zbl 0562.06002
The numbers 1, 2 and 3 are pairwise incomparable points in a poset on $$n\geq 3$$ points. Let N(ijk) be the number of linear extensions of the poset in which i precedes j and j precedes k. On this base $$\lambda$$ is defined by a special formula. Two applications of the Ahlswede-Daykin evaluation theorem for distributive lattices are used to prove that $$\lambda \leq (n-1)^ 2/(n+1)^ 2$$ for odd n, and $$\lambda \leq (n- 2)/(n+2)$$ for even n.
L. A. Shepp [Ann. Probab. 10, 824-827 (1982; Zbl 0484.60010)] proved that P(12)$$\leq P(12| 13)$$, the so-called xyz inequality, where P(ij) is the probability that i precedes j in a randomly chosen linear extension of the poset, thus settling a conjecture of Rival and Sands. The preceding bounds on $$\lambda$$ yield a simple proof of $$P(12)<P(12| 13)$$, which had also been conjectured by Rival and Sands.
Reviewer: F.V.Burshtein

##### MSC:
 06A06 Partial orders, general 05A20 Combinatorial inequalities 60C05 Combinatorial probability
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##### References:
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