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SIAM Journal on Algebraic Discrete Methods
Let (a1 , . . . , am, b1, . . . , bn) be a random permutation of 1, 2, . . ., m + n. Let P be a partial order on the a’s and b’s involving only inequalities of the form ai < aj or bi < bj, and let P' be an extension of P to include inequalities of the form ai < bj; i.e, P' = P ∪ P'', where P'' involves only inequalities of the form ai < bj. We prove the natural conjecture of R. L. Graham, A. C. Yao, and F. F. Yao [SIAM J. Alg. Discr. Meth. 1 (1980), pp. 251–258] that in particular (*) Pr (a1 < b1|P') ≥ Pr (a1 < b1|P). We give a simple example to show that the more general inequality (*) where P is allowed to contain inequalities of the form ai < bj is false. This is surprising because as Graham, Yao, and Yao proved, the general inequality (*) does hold if P totally orders both the a’s and the b’s separately. We give a new proof of the latter result. Our proofs are based on the FKG inequality.
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Shepp, L. A. (1980). The FKG Inequality and Some Monotonicity Properties of Partial Orders. SIAM Journal on Algebraic Discrete Methods, 1 (3), 295-299. http://dx.doi.org/10.1137/0601034
Date Posted: 27 November 2017