Multivariate correlation inequalities for $P$-partitions

TL;DR

This paper introduces broad multivariate inequalities related to P-partitions, extending classical results like Fishburn's and Daykin--Daykin--Paterson's inequalities, with proofs based on a multivariate Ahlswede--Daykin inequality.

Contribution

It provides new multivariate generalizations of key inequalities for P-partitions and poset order polynomials, expanding the theoretical framework in combinatorics.

Findings

Multivariate generalization of Fishburn's correlation inequality.

Multivariate log-concavity of the order polynomial of a poset.

Multivariate P-partition version of the cross-product inequality.

Abstract

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.