Multivariate correlation inequalities for $P$-partitions

Motivated by the Lam–Pylyavskyy inequalities for Schur functions, we give a far reaching multivariate generalization of Fishburn’s correlation inequality for the number of linear extensions of posets. We then give a multivariate generalization of the Daykin–Daykin–Paterson inequality proving log-concavity of the order polynomial of a poset. We also prove a multivariate $P$-partition version of the cross-product inequality by Brightwell–Felsner–Trotter. The proofs are based on a multivariate generalization of the Ahlswede–Daykin inequality.