In combinatorial mathematics, the XYZ inequality, also called the Fishburn–Shepp inequality, is an inequality for the number of linear extensions of finite partial orders.
The inequality was conjectured by Ivan Rival and Bill Sands in 1981.
It states that if x, y, and z are incomparable elements of a finite poset, then where P(A) is the probability that a linear order extending the partial order
increases if one adds the condition that
In the language of conditional probability, The proof uses the Ahlswede–Daykin inequality.