# Generalized semimodularity: order statistics

## TL;DR

This paper introduces a generalized notion of semimodularity extending classical concepts, proving a key inequality for functions on distributive lattices and demonstrating applications in combinatorics, matrix theory, and probability.

## Contribution

It defines generalized n-semimodularity, proves a main inequality extending Muirhead's lemma, and applies results to combinatorics, matrix functions, and correlation inequalities.

## Key findings

- Proves generalized (n:2)-semimodularity implies n-semimodularity on distributive lattices.
- Establishes applications to permanents, symmetric functions, and order statistics.
- Extends FKG inequality for correlation bounds in order statistics.

## Abstract

A notion of generalized $n$-semimodularity is introduced, which extends that of (sub/super)mod\-ularity in four ways at once. The main result of this paper, stating that every generalized $(n\colon\!2)$-semimodular function on the $n$th Cartesian power of a distributive lattice is generalized $n$-semimodular, may be considered a multi/infinite-dimensional analogue of the well-known Muirhead lemma in the theory of Schur majorization. This result is also similar to a discretized version of the well-known theorem due to Lorentz, which latter was given only for additive-type functions. Illustrations of our main result are presented for counts of combinations of faces of a polytope; one-sided potentials; multiadditive forms, including multilinear ones -- in particular, permanents of rectangular matrices and elementary symmetric functions; and association inequalities for order statistics. Based on an extension of the FKG inequality due to Rinott \& Saks and Aharoni \& Keich, applications to correlation inequalities for order statistics are given as well.