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Multivariate correlation inequalities for $P$-partitions

Swee Hong Chan and Igor Pak

Vol. 323 (2023), No. 2, 223–252
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
Authors
Swee Hong Chan
Department of Mathematics
Rutgers University
Piscataway, NJ
United States
Igor Pak
Department of Mathematics
University of California, Los Angeles
Los Angeles, CA
United States

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