Multivariate correlation inequalities for $P$-partitions

Swee Hong Chan and Igor Pak

Vol. 323 (2023), No. 2, 223–252

DOI: 10.2140/pjm.2023.323.223

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 and Trotter. The proofs are based on a multivariate generalization of the Ahlswede–Daykin inequality.

Keywords

poset, linear extension, order polynomial, Schur function, $q$-analogue, log-concavity, Young diagram, $P$-partition, correlation inequality, FKG inequality, Ahlswede–Daykin inequality, XYZ inequality, Daykin–Paterson–Paterson inequality, Lam–Pylyavskyy inequality, Fishburn inequality

Mathematical Subject Classification

Primary: 05A20

Secondary: 05A30, 05E05, 05E10, 06A07, 60C05

Milestones

Received: 6 January 2023

Revised: 23 May 2023

Accepted: 23 May 2023

Published: 2 June 2023