Log-concavity and log-convexity via distributive lattices

TL;DR

This paper introduces the Order Ideal Lemma, a combinatorial tool for proving log-concavity and log-convexity across various sequences and structures using distributive lattices.

Contribution

The paper presents the Order Ideal Lemma and demonstrates its application to establish log-concavity and log-convexity in diverse combinatorial sequences and objects.

Findings

Proves log-concavity of Catalan, Motzkin, and Schröder numbers.

Establishes log-convexity of certain sequences in Young's lattice and Schur functions.

Provides conjectures and future research directions.

Abstract

We prove a lemma, which we call the Order Ideal Lemma, that can be used to demonstrate a wide array of log-concavity and log-convexity results in a combinatorial manner using order ideals in distributive lattices. We use the Order Ideal Lemma to prove log-concavity and log-convexity of various sequences involving lattice paths (Catalan, Motzkin and large Schr\"oder numbers), intervals in Young's lattice, order polynomials, specializations of Schur and Schur Q-functions, Lucas sequences, descent and peak polynomials of permutations, pattern avoidance, set partitions, and noncrossing partitions. We end with a section with conjectures and outlining future directions.